| L(s) = 1 | + 2-s + 2.83·3-s + 4-s + 2.83·6-s + 3.62·7-s + 8-s + 5.05·9-s + 2.83·12-s + 3.62·14-s + 16-s − 1.16·17-s + 5.05·18-s − 19-s + 10.2·21-s + 7.73·23-s + 2.83·24-s − 5·25-s + 5.83·27-s + 3.62·28-s + 4.45·29-s − 0.436·31-s + 32-s − 1.16·34-s + 5.05·36-s + 10.6·37-s − 38-s − 11.7·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.63·3-s + 0.5·4-s + 1.15·6-s + 1.36·7-s + 0.353·8-s + 1.68·9-s + 0.819·12-s + 0.967·14-s + 0.250·16-s − 0.281·17-s + 1.19·18-s − 0.229·19-s + 2.24·21-s + 1.61·23-s + 0.579·24-s − 25-s + 1.12·27-s + 0.684·28-s + 0.827·29-s − 0.0784·31-s + 0.176·32-s − 0.199·34-s + 0.842·36-s + 1.75·37-s − 0.162·38-s − 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.350009322\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.350009322\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.83T + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 3.62T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 1.16T + 17T^{2} \) |
| 23 | \( 1 - 7.73T + 23T^{2} \) |
| 29 | \( 1 - 4.45T + 29T^{2} \) |
| 31 | \( 1 + 0.436T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 11.7T + 41T^{2} \) |
| 43 | \( 1 + 1.67T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 2.89T + 53T^{2} \) |
| 59 | \( 1 - 6.78T + 59T^{2} \) |
| 61 | \( 1 + 0.323T + 61T^{2} \) |
| 67 | \( 1 + 1.21T + 67T^{2} \) |
| 71 | \( 1 + 14.1T + 71T^{2} \) |
| 73 | \( 1 + 14.1T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 8.11T + 83T^{2} \) |
| 89 | \( 1 + 2.87T + 89T^{2} \) |
| 97 | \( 1 - 1.67T + 97T^{2} \) |
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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
−8.183840584670173156955476321237, −7.40416757967010540869554178726, −6.82335040663558580870616366483, −5.80275284468523957972975449055, −4.74471690059360123260897169159, −4.51252054262508051622177499366, −3.48864103428193659018033534895, −2.86110117442313747967282543296, −2.02777701164443732505189374764, −1.35833857109268087718193801191,
1.35833857109268087718193801191, 2.02777701164443732505189374764, 2.86110117442313747967282543296, 3.48864103428193659018033534895, 4.51252054262508051622177499366, 4.74471690059360123260897169159, 5.80275284468523957972975449055, 6.82335040663558580870616366483, 7.40416757967010540869554178726, 8.183840584670173156955476321237
