| L(s) = 1 | + 1.43·2-s + 1.95·3-s + 0.0674·4-s − 2.13·5-s + 2.81·6-s − 3.51·7-s − 2.77·8-s + 0.827·9-s − 3.06·10-s + 0.131·12-s − 4.22·13-s − 5.05·14-s − 4.17·15-s − 4.13·16-s + 6.93·17-s + 1.18·18-s − 4.37·19-s − 0.143·20-s − 6.88·21-s + 3.18·23-s − 5.43·24-s − 0.452·25-s − 6.07·26-s − 4.25·27-s − 0.237·28-s + 8.09·29-s − 5.99·30-s + ⋯ |
| L(s) = 1 | + 1.01·2-s + 1.12·3-s + 0.0337·4-s − 0.953·5-s + 1.14·6-s − 1.32·7-s − 0.982·8-s + 0.275·9-s − 0.969·10-s + 0.0381·12-s − 1.17·13-s − 1.35·14-s − 1.07·15-s − 1.03·16-s + 1.68·17-s + 0.280·18-s − 1.00·19-s − 0.0321·20-s − 1.50·21-s + 0.663·23-s − 1.10·24-s − 0.0905·25-s − 1.19·26-s − 0.818·27-s − 0.0448·28-s + 1.50·29-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.139902414\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.139902414\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 1.43T + 2T^{2} \) |
| 3 | \( 1 - 1.95T + 3T^{2} \) |
| 5 | \( 1 + 2.13T + 5T^{2} \) |
| 7 | \( 1 + 3.51T + 7T^{2} \) |
| 13 | \( 1 + 4.22T + 13T^{2} \) |
| 17 | \( 1 - 6.93T + 17T^{2} \) |
| 19 | \( 1 + 4.37T + 19T^{2} \) |
| 23 | \( 1 - 3.18T + 23T^{2} \) |
| 29 | \( 1 - 8.09T + 29T^{2} \) |
| 31 | \( 1 - 2.88T + 31T^{2} \) |
| 37 | \( 1 - 8.14T + 37T^{2} \) |
| 41 | \( 1 - 4.27T + 41T^{2} \) |
| 43 | \( 1 - 7.99T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 3.87T + 53T^{2} \) |
| 59 | \( 1 + 7.58T + 59T^{2} \) |
| 67 | \( 1 - 5.08T + 67T^{2} \) |
| 71 | \( 1 + 2.96T + 71T^{2} \) |
| 73 | \( 1 + 7.70T + 73T^{2} \) |
| 79 | \( 1 - 4.13T + 79T^{2} \) |
| 83 | \( 1 - 9.72T + 83T^{2} \) |
| 89 | \( 1 - 1.48T + 89T^{2} \) |
| 97 | \( 1 + 1.99T + 97T^{2} \) |
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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
−7.943251543763533633067038458611, −7.26837169737372657194506415253, −6.38604892225328768717764694658, −5.82977888378779840740029571929, −4.75333501706367721030604366797, −4.26506428351573428211919562960, −3.30815382200881325981350190688, −3.14227437281911926728314998411, −2.41945318073973671025689691345, −0.56182583786477401584968921514,
0.56182583786477401584968921514, 2.41945318073973671025689691345, 3.14227437281911926728314998411, 3.30815382200881325981350190688, 4.26506428351573428211919562960, 4.75333501706367721030604366797, 5.82977888378779840740029571929, 6.38604892225328768717764694658, 7.26837169737372657194506415253, 7.943251543763533633067038458611
