| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s − 7-s + 9-s + 4·11-s − 2·12-s + 13-s − 2·14-s − 4·16-s + 2·18-s + 5·19-s + 21-s + 8·22-s + 2·23-s + 2·26-s − 27-s − 2·28-s − 6·29-s − 3·31-s − 8·32-s − 4·33-s + 2·36-s + 37-s + 10·38-s − 39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.577·12-s + 0.277·13-s − 0.534·14-s − 16-s + 0.471·18-s + 1.14·19-s + 0.218·21-s + 1.70·22-s + 0.417·23-s + 0.392·26-s − 0.192·27-s − 0.377·28-s − 1.11·29-s − 0.538·31-s − 1.41·32-s − 0.696·33-s + 1/3·36-s + 0.164·37-s + 1.62·38-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.640884582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.640884582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 11 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.23887358957367, −12.88817213461001, −12.37281067457870, −12.11846148754582, −11.41414137553963, −11.26847135084209, −10.77456749656133, −10.00097619954092, −9.359452647321633, −9.208403175777890, −8.637917031159939, −7.570909182203251, −7.380932734742995, −6.740472668085025, −6.229257400492520, −5.805114541099773, −5.481075015928804, −4.805178950207031, −4.250937620950321, −3.854878675853792, −3.333688451622098, −2.792919223268695, −1.989737634631714, −1.263241285861579, −0.5389327647271995,
0.5389327647271995, 1.263241285861579, 1.989737634631714, 2.792919223268695, 3.333688451622098, 3.854878675853792, 4.250937620950321, 4.805178950207031, 5.481075015928804, 5.805114541099773, 6.229257400492520, 6.740472668085025, 7.380932734742995, 7.570909182203251, 8.637917031159939, 9.208403175777890, 9.359452647321633, 10.00097619954092, 10.77456749656133, 11.26847135084209, 11.41414137553963, 12.11846148754582, 12.37281067457870, 12.88817213461001, 13.23887358957367
