| L(s) = 1 | − 3·9-s + 6·11-s + 8·17-s − 3·19-s − 7·25-s + 7·41-s − 14·43-s + 2·49-s + 15·59-s + 5·67-s − 5·73-s + 9·81-s + 10·83-s − 2·89-s − 7·97-s − 18·99-s + 8·107-s + 18·113-s + 9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 24·153-s + 157-s + ⋯ |
| L(s) = 1 | − 9-s + 1.80·11-s + 1.94·17-s − 0.688·19-s − 7/5·25-s + 1.09·41-s − 2.13·43-s + 2/7·49-s + 1.95·59-s + 0.610·67-s − 0.585·73-s + 81-s + 1.09·83-s − 0.211·89-s − 0.710·97-s − 1.80·99-s + 0.773·107-s + 1.69·113-s + 9/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s − 1.94·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 831744 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.027221978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.027221978\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p T^{2} \) |
| 19 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 17 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 42 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 33 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 81 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.167382874845756993955112624663, −7.900931778637630404335778295083, −7.38010545605440214746415397919, −6.79897578911475174145227651458, −6.39927656720722689192970932015, −6.04239878563227784881767341841, −5.46530275662544655086673434383, −5.26674606631096123941954863958, −4.38966538639071214667679547246, −3.91638952287216343352342891818, −3.52780864025982773961006148588, −3.03397147794067456137657120929, −2.18761676304028509887413395776, −1.56821416988438493963816088949, −0.71108666416472083269724855377,
0.71108666416472083269724855377, 1.56821416988438493963816088949, 2.18761676304028509887413395776, 3.03397147794067456137657120929, 3.52780864025982773961006148588, 3.91638952287216343352342891818, 4.38966538639071214667679547246, 5.26674606631096123941954863958, 5.46530275662544655086673434383, 6.04239878563227784881767341841, 6.39927656720722689192970932015, 6.79897578911475174145227651458, 7.38010545605440214746415397919, 7.900931778637630404335778295083, 8.167382874845756993955112624663
