| L(s) = 1 | − 3·3-s + 21.0·5-s − 7·7-s + 9·9-s + 59.9·11-s + 58.8·13-s − 63.2·15-s + 56.5·17-s + 17.6·19-s + 21·21-s + 140.·23-s + 319.·25-s − 27·27-s − 203.·29-s − 215.·31-s − 179.·33-s − 147.·35-s + 265.·37-s − 176.·39-s + 256.·41-s − 119.·43-s + 189.·45-s + 118.·47-s + 49·49-s − 169.·51-s − 106.·53-s + 1.26e3·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.88·5-s − 0.377·7-s + 0.333·9-s + 1.64·11-s + 1.25·13-s − 1.08·15-s + 0.806·17-s + 0.212·19-s + 0.218·21-s + 1.27·23-s + 2.55·25-s − 0.192·27-s − 1.30·29-s − 1.24·31-s − 0.948·33-s − 0.712·35-s + 1.17·37-s − 0.725·39-s + 0.975·41-s − 0.423·43-s + 0.628·45-s + 0.366·47-s + 0.142·49-s − 0.465·51-s − 0.277·53-s + 3.09·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.346064670\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.346064670\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 21.0T + 125T^{2} \) |
| 11 | \( 1 - 59.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 58.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 56.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 17.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 203.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 215.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 265.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 256.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 119.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 118.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 106.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 846.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 875.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 530.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 629.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 843.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 332.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 525.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 362.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 272.T + 9.12e5T^{2} \) |
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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
−9.261168050042529704699156142017, −8.909827839896148693168491234531, −7.32802857375040382745282652111, −6.49691726380090003473607106729, −5.93554707718714531656307882591, −5.41608956515903586054582002292, −4.11175205926162876070431757936, −3.05681624553291415376237580919, −1.61893608593750226114150517798, −1.09756988350333434088323477837,
1.09756988350333434088323477837, 1.61893608593750226114150517798, 3.05681624553291415376237580919, 4.11175205926162876070431757936, 5.41608956515903586054582002292, 5.93554707718714531656307882591, 6.49691726380090003473607106729, 7.32802857375040382745282652111, 8.909827839896148693168491234531, 9.261168050042529704699156142017
