| L(s) = 1 | − 394·3-s + 2.04e3·4-s + 1.77e5·9-s − 4.25e5·11-s − 8.06e5·12-s + 2.42e7·19-s + 9.76e7·25-s − 1.48e8·27-s + 1.67e8·33-s + 3.62e8·36-s + 3.53e9·41-s − 2.19e8·43-s − 8.71e8·44-s − 3.95e9·49-s − 9.53e9·57-s + 3.14e10·59-s − 8.58e9·64-s + 2.11e10·67-s + 6.80e10·73-s − 3.84e10·75-s + 4.95e10·76-s + 5.84e10·81-s − 6.98e10·97-s − 7.54e10·99-s + 2.00e11·100-s − 3.03e11·108-s − 1.84e11·121-s + ⋯ |
| L(s) = 1 | − 0.936·3-s + 4-s + 9-s − 0.797·11-s − 0.936·12-s + 2.24·19-s + 2·25-s − 1.98·27-s + 0.746·33-s + 36-s + 4.77·41-s − 0.227·43-s − 0.797·44-s − 2·49-s − 2.09·57-s + 5.73·59-s − 64-s + 1.91·67-s + 3.84·73-s − 1.87·75-s + 2.24·76-s + 1.86·81-s − 0.826·97-s − 0.797·99-s + 2·100-s − 1.98·108-s − 0.647·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+11/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(\approx\) |
\(7.051211689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.051211689\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 - p^{11} T^{2} + p^{22} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 394 T - 21911 T^{2} + 394 p^{11} T^{3} + p^{22} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + 141906 T + p^{11} T^{2} )^{2}( 1 + 141906 T - 265174357775 T^{2} + 141906 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 13 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 17 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 5215578 T - 7069642433549 T^{2} - 5215578 p^{11} T^{3} + p^{22} T^{4} )( 1 + 5215578 T - 7069642433549 T^{2} + 5215578 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - 12105322 T + 30048561825465 T^{2} - 12105322 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p^{11} T^{2} )^{4} \) |
| 41 | $C_2$$\times$$C_2^2$ | \( ( 1 - 1179887286 T + p^{11} T^{2} )^{2}( 1 - 1179887286 T + 841804975948197355 T^{2} - 1179887286 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 43 | $C_2$$\times$$C_2^2$ | \( ( 1 + 219226270 T + p^{11} T^{2} )^{2}( 1 - 219226270 T - 881233582013109807 T^{2} - 219226270 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 47 | $C_2^2$ | \( ( 1 - p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{4} \) |
| 59 | $C_2$$\times$$C_2^2$ | \( ( 1 - 10496647794 T + p^{11} T^{2} )^{2}( 1 - 10496647794 T + 80023726466547223777 T^{2} - 10496647794 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 61 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 - 21183162938 T + p^{11} T^{2} )^{2}( 1 + 21183162938 T + \)\(32\!\cdots\!61\)\( T^{2} + 21183162938 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p^{11} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 - 34041283054 T + \)\(84\!\cdots\!39\)\( T^{2} - 34041283054 p^{11} T^{3} + p^{22} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + p^{11} T^{2} + p^{22} T^{4} )^{2} \) |
| 83 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 10275997626 T - \)\(11\!\cdots\!91\)\( T^{2} - 10275997626 p^{11} T^{3} + p^{22} T^{4} )( 1 + 10275997626 T - \)\(11\!\cdots\!91\)\( T^{2} + 10275997626 p^{11} T^{3} + p^{22} T^{4} ) \) |
| 89 | $C_2$ | \( ( 1 - 103091787198 T + p^{11} T^{2} )^{2}( 1 + 103091787198 T + p^{11} T^{2} )^{2} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 69880926730 T + p^{11} T^{2} )^{2}( 1 - 69880926730 T - \)\(22\!\cdots\!53\)\( T^{2} - 69880926730 p^{11} T^{3} + p^{22} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.450620020967459602933768965206, −8.131587613023812599429779682955, −7.73382938805541796859003531909, −7.47754390108269800105006897815, −7.21548108815147635927436822670, −7.00024580395537160184432364533, −6.71196493588632390155744751968, −6.26592772938810651879233381075, −6.08742262591700190883056473230, −5.62149641610070985309824690694, −5.19724289448914034838904023291, −5.18253810845594788321792761642, −4.95115601931734077150493954217, −4.24487991454872261748376249059, −4.00749869261657194721924279001, −3.52458165896364461972413493144, −3.36241067195624579222507339519, −2.54662983770227832781234333940, −2.48978642370324552150620129501, −2.34780621771631867577252369284, −1.74604013202517595204412132850, −1.10998041922420653523693602560, −0.873207805342512443187321830952, −0.836331778228474756298553764522, −0.34547410781147275572142052274,
0.34547410781147275572142052274, 0.836331778228474756298553764522, 0.873207805342512443187321830952, 1.10998041922420653523693602560, 1.74604013202517595204412132850, 2.34780621771631867577252369284, 2.48978642370324552150620129501, 2.54662983770227832781234333940, 3.36241067195624579222507339519, 3.52458165896364461972413493144, 4.00749869261657194721924279001, 4.24487991454872261748376249059, 4.95115601931734077150493954217, 5.18253810845594788321792761642, 5.19724289448914034838904023291, 5.62149641610070985309824690694, 6.08742262591700190883056473230, 6.26592772938810651879233381075, 6.71196493588632390155744751968, 7.00024580395537160184432364533, 7.21548108815147635927436822670, 7.47754390108269800105006897815, 7.73382938805541796859003531909, 8.131587613023812599429779682955, 8.450620020967459602933768965206
