| L(s) = 1 | + 2-s + 4-s + 8-s + 6·11-s − 13-s + 16-s + 7·17-s − 2·19-s + 6·22-s + 3·23-s − 26-s + 6·29-s − 10·31-s + 32-s + 7·34-s + 6·37-s − 2·38-s + 2·41-s − 5·43-s + 6·44-s + 3·46-s + 12·47-s − 7·49-s − 52-s − 4·53-s + 6·58-s + 9·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.80·11-s − 0.277·13-s + 1/4·16-s + 1.69·17-s − 0.458·19-s + 1.27·22-s + 0.625·23-s − 0.196·26-s + 1.11·29-s − 1.79·31-s + 0.176·32-s + 1.20·34-s + 0.986·37-s − 0.324·38-s + 0.312·41-s − 0.762·43-s + 0.904·44-s + 0.442·46-s + 1.75·47-s − 49-s − 0.138·52-s − 0.549·53-s + 0.787·58-s + 1.17·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.686209546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.686209546\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 127 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.28983485908599, −14.21591457730817, −13.30054145112136, −12.76226754356548, −12.40488874908108, −11.89744964262236, −11.42522963160634, −10.99316731643731, −10.27282279164121, −9.717467621038128, −9.326546394399644, −8.637987239704574, −8.104934868129082, −7.383101933274070, −6.920645120161136, −6.451948168289956, −5.783467614848795, −5.348574231380920, −4.643486276257176, −3.970711246905709, −3.610128500637639, −2.947383929924249, −2.144681136897722, −1.356222906816163, −0.7968306489888861,
0.7968306489888861, 1.356222906816163, 2.144681136897722, 2.947383929924249, 3.610128500637639, 3.970711246905709, 4.643486276257176, 5.348574231380920, 5.783467614848795, 6.451948168289956, 6.920645120161136, 7.383101933274070, 8.104934868129082, 8.637987239704574, 9.326546394399644, 9.717467621038128, 10.27282279164121, 10.99316731643731, 11.42522963160634, 11.89744964262236, 12.40488874908108, 12.76226754356548, 13.30054145112136, 14.21591457730817, 14.28983485908599
