| L(s) = 1 | + 3·3-s − 2·4-s − 5-s + 7-s + 6·9-s + 2·11-s − 6·12-s + 4·13-s − 3·15-s + 4·16-s + 6·17-s − 5·19-s + 2·20-s + 3·21-s + 3·23-s + 25-s + 9·27-s − 2·28-s − 4·31-s + 6·33-s − 35-s − 12·36-s + 6·37-s + 12·39-s + 7·41-s − 6·43-s − 4·44-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 4-s − 0.447·5-s + 0.377·7-s + 2·9-s + 0.603·11-s − 1.73·12-s + 1.10·13-s − 0.774·15-s + 16-s + 1.45·17-s − 1.14·19-s + 0.447·20-s + 0.654·21-s + 0.625·23-s + 1/5·25-s + 1.73·27-s − 0.377·28-s − 0.718·31-s + 1.04·33-s − 0.169·35-s − 2·36-s + 0.986·37-s + 1.92·39-s + 1.09·41-s − 0.914·43-s − 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.629538448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.629538448\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 13 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−14.97860810719517, −14.49842603134345, −14.22065182390354, −13.72346422152449, −13.12695176525105, −12.68911716881026, −12.27234213806963, −11.34839736597195, −10.75829071089381, −10.15666920186417, −9.433564977828105, −9.136551070372590, −8.616732093520846, −8.100314980141101, −7.820173196343585, −7.132661105659263, −6.298619521926641, −5.550296421143570, −4.746839753831632, −3.992426592686086, −3.814515258497473, −3.191392430149667, −2.373947889341594, −1.424316992801923, −0.8506407510734369,
0.8506407510734369, 1.424316992801923, 2.373947889341594, 3.191392430149667, 3.814515258497473, 3.992426592686086, 4.746839753831632, 5.550296421143570, 6.298619521926641, 7.132661105659263, 7.820173196343585, 8.100314980141101, 8.616732093520846, 9.136551070372590, 9.433564977828105, 10.15666920186417, 10.75829071089381, 11.34839736597195, 12.27234213806963, 12.68911716881026, 13.12695176525105, 13.72346422152449, 14.22065182390354, 14.49842603134345, 14.97860810719517
