L(s) = 1 | + 2.37·2-s + 3.64·4-s − 0.0257·5-s + 0.278·7-s + 3.91·8-s − 0.0612·10-s + 2.03·11-s − 5.15·13-s + 0.662·14-s + 2.01·16-s + 6.10·17-s + 7.62·19-s − 0.0940·20-s + 4.84·22-s + 9.14·23-s − 4.99·25-s − 12.2·26-s + 1.01·28-s + 2.35·29-s + 2.04·31-s − 3.04·32-s + 14.5·34-s − 0.00718·35-s − 1.68·37-s + 18.1·38-s − 0.100·40-s + 5.01·41-s + ⋯ |
L(s) = 1 | + 1.68·2-s + 1.82·4-s − 0.0115·5-s + 0.105·7-s + 1.38·8-s − 0.0193·10-s + 0.614·11-s − 1.42·13-s + 0.177·14-s + 0.503·16-s + 1.48·17-s + 1.74·19-s − 0.0210·20-s + 1.03·22-s + 1.90·23-s − 0.999·25-s − 2.40·26-s + 0.192·28-s + 0.436·29-s + 0.367·31-s − 0.538·32-s + 2.48·34-s − 0.00121·35-s − 0.276·37-s + 2.93·38-s − 0.0159·40-s + 0.783·41-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
\(L(1)\) |
\(\approx\) |
\(5.133152679\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.133152679\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.37T + 2T^{2} \) |
| 5 | \( 1 + 0.0257T + 5T^{2} \) |
| 7 | \( 1 - 0.278T + 7T^{2} \) |
| 11 | \( 1 - 2.03T + 11T^{2} \) |
| 13 | \( 1 + 5.15T + 13T^{2} \) |
| 17 | \( 1 - 6.10T + 17T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 23 | \( 1 - 9.14T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 - 2.04T + 31T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + 6.24T + 47T^{2} \) |
| 53 | \( 1 + 7.02T + 53T^{2} \) |
| 59 | \( 1 + 0.590T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 13.1T + 71T^{2} \) |
| 73 | \( 1 + 15.7T + 73T^{2} \) |
| 79 | \( 1 + 0.773T + 79T^{2} \) |
| 83 | \( 1 - 0.895T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 + 7.22T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.356199247428263749092082477740, −7.902010047540336589879644903203, −7.29810509889904469682375869539, −6.60281785613068289736185620155, −5.51229944262052364945004570728, −5.16799413589446436531567500040, −4.29999304384813745069525379792, −3.27402316911641604685738190629, −2.75603769511792547114373413992, −1.31142384981614084987799661220,
1.31142384981614084987799661220, 2.75603769511792547114373413992, 3.27402316911641604685738190629, 4.29999304384813745069525379792, 5.16799413589446436531567500040, 5.51229944262052364945004570728, 6.60281785613068289736185620155, 7.29810509889904469682375869539, 7.902010047540336589879644903203, 9.356199247428263749092082477740