| L(s) = 1 | + 2-s + 1.53·3-s + 4-s + 1.53·6-s + 4.67·7-s + 8-s − 0.653·9-s + 0.0451·11-s + 1.53·12-s + 5.26·13-s + 4.67·14-s + 16-s + 3.60·17-s − 0.653·18-s − 6.22·19-s + 7.15·21-s + 0.0451·22-s − 2.20·23-s + 1.53·24-s + 5.26·26-s − 5.59·27-s + 4.67·28-s − 4.20·29-s − 3.01·31-s + 32-s + 0.0691·33-s + 3.60·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.884·3-s + 0.5·4-s + 0.625·6-s + 1.76·7-s + 0.353·8-s − 0.217·9-s + 0.0136·11-s + 0.442·12-s + 1.46·13-s + 1.24·14-s + 0.250·16-s + 0.875·17-s − 0.154·18-s − 1.42·19-s + 1.56·21-s + 0.00962·22-s − 0.459·23-s + 0.312·24-s + 1.03·26-s − 1.07·27-s + 0.882·28-s − 0.780·29-s − 0.540·31-s + 0.176·32-s + 0.0120·33-s + 0.618·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.430170398\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.430170398\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 1.53T + 3T^{2} \) |
| 7 | \( 1 - 4.67T + 7T^{2} \) |
| 11 | \( 1 - 0.0451T + 11T^{2} \) |
| 13 | \( 1 - 5.26T + 13T^{2} \) |
| 17 | \( 1 - 3.60T + 17T^{2} \) |
| 19 | \( 1 + 6.22T + 19T^{2} \) |
| 23 | \( 1 + 2.20T + 23T^{2} \) |
| 29 | \( 1 + 4.20T + 29T^{2} \) |
| 31 | \( 1 + 3.01T + 31T^{2} \) |
| 41 | \( 1 + 7.38T + 41T^{2} \) |
| 43 | \( 1 + 5.54T + 43T^{2} \) |
| 47 | \( 1 + 4.28T + 47T^{2} \) |
| 53 | \( 1 - 6.10T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 4.27T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 4.49T + 71T^{2} \) |
| 73 | \( 1 - 7.03T + 73T^{2} \) |
| 79 | \( 1 + 8.72T + 79T^{2} \) |
| 83 | \( 1 + 0.880T + 83T^{2} \) |
| 89 | \( 1 - 9.97T + 89T^{2} \) |
| 97 | \( 1 - 0.240T + 97T^{2} \) |
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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
−8.904352806703044401284728677614, −8.262584487460450375114082737507, −8.004977542515655610760039337312, −6.88411339106705217998482809362, −5.82270825649933118868813829167, −5.18765671103623640760396080630, −4.09390755780666900007470533136, −3.53452258516760280778564800963, −2.24745682026215549323647861733, −1.52322316566165997344054467393,
1.52322316566165997344054467393, 2.24745682026215549323647861733, 3.53452258516760280778564800963, 4.09390755780666900007470533136, 5.18765671103623640760396080630, 5.82270825649933118868813829167, 6.88411339106705217998482809362, 8.004977542515655610760039337312, 8.262584487460450375114082737507, 8.904352806703044401284728677614
