| L(s) = 1 | − 1.86·2-s + 1.47·4-s + 3.80·5-s + 4.40·7-s + 0.980·8-s − 7.08·10-s − 5.43·11-s + 0.253·13-s − 8.20·14-s − 4.77·16-s − 5.22·17-s + 5.87·19-s + 5.60·20-s + 10.1·22-s + 4.24·23-s + 9.44·25-s − 0.471·26-s + 6.48·28-s − 6.77·29-s − 6.72·31-s + 6.94·32-s + 9.74·34-s + 16.7·35-s + 8.64·37-s − 10.9·38-s + 3.72·40-s + 6.28·41-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.737·4-s + 1.69·5-s + 1.66·7-s + 0.346·8-s − 2.23·10-s − 1.64·11-s + 0.0702·13-s − 2.19·14-s − 1.19·16-s − 1.26·17-s + 1.34·19-s + 1.25·20-s + 2.16·22-s + 0.885·23-s + 1.88·25-s − 0.0925·26-s + 1.22·28-s − 1.25·29-s − 1.20·31-s + 1.22·32-s + 1.67·34-s + 2.82·35-s + 1.42·37-s − 1.77·38-s + 0.588·40-s + 0.980·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.396350502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.396350502\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 + T \) |
| good | 2 | \( 1 + 1.86T + 2T^{2} \) |
| 5 | \( 1 - 3.80T + 5T^{2} \) |
| 7 | \( 1 - 4.40T + 7T^{2} \) |
| 11 | \( 1 + 5.43T + 11T^{2} \) |
| 13 | \( 1 - 0.253T + 13T^{2} \) |
| 17 | \( 1 + 5.22T + 17T^{2} \) |
| 19 | \( 1 - 5.87T + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 + 6.77T + 29T^{2} \) |
| 31 | \( 1 + 6.72T + 31T^{2} \) |
| 37 | \( 1 - 8.64T + 37T^{2} \) |
| 41 | \( 1 - 6.28T + 41T^{2} \) |
| 43 | \( 1 - 3.72T + 43T^{2} \) |
| 47 | \( 1 - 8.12T + 47T^{2} \) |
| 53 | \( 1 - 6.50T + 53T^{2} \) |
| 59 | \( 1 - 2.21T + 59T^{2} \) |
| 61 | \( 1 + 1.92T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 + 7.92T + 71T^{2} \) |
| 73 | \( 1 - 1.11T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 3.28T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 6.15T + 97T^{2} \) |
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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
−9.143195780727482329080710161041, −8.493548712761788925378634204720, −7.56314749052015133354031125149, −7.23950777869554946128797981143, −5.76746518654772521936211849905, −5.29274836989578276010559928953, −4.50782698752062643779499593980, −2.50954484424773842606958206760, −2.01188395515728202827994106607, −1.00069469978187152852826015441,
1.00069469978187152852826015441, 2.01188395515728202827994106607, 2.50954484424773842606958206760, 4.50782698752062643779499593980, 5.29274836989578276010559928953, 5.76746518654772521936211849905, 7.23950777869554946128797981143, 7.56314749052015133354031125149, 8.493548712761788925378634204720, 9.143195780727482329080710161041
