L(s) = 1 | − 4-s + 2·5-s + 6·9-s − 6·11-s + 16-s − 2·20-s − 25-s + 6·29-s − 14·31-s − 6·36-s + 22·41-s + 6·44-s + 12·45-s + 13·49-s − 12·55-s + 24·59-s − 30·61-s − 64-s + 12·71-s + 16·79-s + 2·80-s + 27·81-s − 36·99-s + 100-s + 28·101-s + 22·109-s − 6·116-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 0.894·5-s + 2·9-s − 1.80·11-s + 1/4·16-s − 0.447·20-s − 1/5·25-s + 1.11·29-s − 2.51·31-s − 36-s + 3.43·41-s + 0.904·44-s + 1.78·45-s + 13/7·49-s − 1.61·55-s + 3.12·59-s − 3.84·61-s − 1/8·64-s + 1.42·71-s + 1.80·79-s + 0.223·80-s + 3·81-s − 3.61·99-s + 1/10·100-s + 2.78·101-s + 2.10·109-s − 0.557·116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.765253783\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.765253783\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 37 | $C_2$ | \( 1 + T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 193 T^{2} + p^{2} T^{4} \) |
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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
−11.63289232736633735697647417301, −10.73736905781209843087631161955, −10.65034694907630538142505323293, −10.39471150879548202412941538275, −9.705363810903186794390782297237, −9.456962441609632349337876712251, −9.070640383729880442900989691839, −8.401752282675722599679890727689, −7.59507897592198116704301180506, −7.53017608381860795614689951682, −7.13819625755538232707465451412, −6.06250635707457948475493231352, −5.99839672215665661936151062504, −5.03291239592663155814992427452, −4.96138051258231117959969714149, −4.08402515918911381702754876324, −3.67096183890071427923843794038, −2.48802850278871505466995445953, −2.10535661502614280226224434512, −0.969120633073486118502555498275,
0.969120633073486118502555498275, 2.10535661502614280226224434512, 2.48802850278871505466995445953, 3.67096183890071427923843794038, 4.08402515918911381702754876324, 4.96138051258231117959969714149, 5.03291239592663155814992427452, 5.99839672215665661936151062504, 6.06250635707457948475493231352, 7.13819625755538232707465451412, 7.53017608381860795614689951682, 7.59507897592198116704301180506, 8.401752282675722599679890727689, 9.070640383729880442900989691839, 9.456962441609632349337876712251, 9.705363810903186794390782297237, 10.39471150879548202412941538275, 10.65034694907630538142505323293, 10.73736905781209843087631161955, 11.63289232736633735697647417301