L(s) = 1 | + 2-s + 1.28·3-s + 4-s + 1.29·5-s + 1.28·6-s + 1.61·7-s + 8-s − 1.35·9-s + 1.29·10-s + 6.44·11-s + 1.28·12-s + 4.80·13-s + 1.61·14-s + 1.66·15-s + 16-s + 4.51·17-s − 1.35·18-s − 3.28·19-s + 1.29·20-s + 2.07·21-s + 6.44·22-s + 1.47·23-s + 1.28·24-s − 3.31·25-s + 4.80·26-s − 5.58·27-s + 1.61·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.740·3-s + 0.5·4-s + 0.580·5-s + 0.523·6-s + 0.611·7-s + 0.353·8-s − 0.452·9-s + 0.410·10-s + 1.94·11-s + 0.370·12-s + 1.33·13-s + 0.432·14-s + 0.429·15-s + 0.250·16-s + 1.09·17-s − 0.319·18-s − 0.753·19-s + 0.290·20-s + 0.452·21-s + 1.37·22-s + 0.306·23-s + 0.261·24-s − 0.663·25-s + 0.943·26-s − 1.07·27-s + 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.985921444\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.985921444\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 + T \) |
good | 3 | \( 1 - 1.28T + 3T^{2} \) |
| 5 | \( 1 - 1.29T + 5T^{2} \) |
| 7 | \( 1 - 1.61T + 7T^{2} \) |
| 11 | \( 1 - 6.44T + 11T^{2} \) |
| 13 | \( 1 - 4.80T + 13T^{2} \) |
| 17 | \( 1 - 4.51T + 17T^{2} \) |
| 19 | \( 1 + 3.28T + 19T^{2} \) |
| 23 | \( 1 - 1.47T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 37 | \( 1 - 0.140T + 37T^{2} \) |
| 41 | \( 1 + 0.752T + 41T^{2} \) |
| 43 | \( 1 + 5.28T + 43T^{2} \) |
| 47 | \( 1 + 0.807T + 47T^{2} \) |
| 53 | \( 1 + 7.00T + 53T^{2} \) |
| 59 | \( 1 + 2.78T + 59T^{2} \) |
| 61 | \( 1 + 2.17T + 61T^{2} \) |
| 67 | \( 1 - 3.44T + 67T^{2} \) |
| 71 | \( 1 - 7.23T + 71T^{2} \) |
| 73 | \( 1 + 9.37T + 73T^{2} \) |
| 79 | \( 1 + 17.3T + 79T^{2} \) |
| 83 | \( 1 - 3.97T + 83T^{2} \) |
| 89 | \( 1 + 2.53T + 89T^{2} \) |
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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
−8.275245899702227858290641666939, −7.32892734636142592634020654060, −6.32589750883817759315845004607, −6.12536082020646332341012194297, −5.22532095592467339511487567974, −4.24148638431508149123250905145, −3.64572062811603354898084773776, −2.99326015556310738724218352412, −1.80250673864584136955980072881, −1.32757200059077158367979457419,
1.32757200059077158367979457419, 1.80250673864584136955980072881, 2.99326015556310738724218352412, 3.64572062811603354898084773776, 4.24148638431508149123250905145, 5.22532095592467339511487567974, 6.12536082020646332341012194297, 6.32589750883817759315845004607, 7.32892734636142592634020654060, 8.275245899702227858290641666939