| L(s) = 1 | − 5.36·2-s − 0.836·3-s + 20.7·4-s + 1.92·5-s + 4.48·6-s + 17.6·7-s − 68.6·8-s − 26.3·9-s − 10.3·10-s − 11·11-s − 17.3·12-s − 94.8·14-s − 1.61·15-s + 201.·16-s − 29.3·17-s + 141.·18-s − 78.8·19-s + 40.0·20-s − 14.7·21-s + 59.0·22-s − 167.·23-s + 57.3·24-s − 121.·25-s + 44.5·27-s + 367.·28-s − 191.·29-s + 8.65·30-s + ⋯ |
| L(s) = 1 | − 1.89·2-s − 0.160·3-s + 2.59·4-s + 0.172·5-s + 0.305·6-s + 0.954·7-s − 3.03·8-s − 0.974·9-s − 0.327·10-s − 0.301·11-s − 0.418·12-s − 1.81·14-s − 0.0277·15-s + 3.15·16-s − 0.418·17-s + 1.84·18-s − 0.951·19-s + 0.448·20-s − 0.153·21-s + 0.571·22-s − 1.52·23-s + 0.488·24-s − 0.970·25-s + 0.317·27-s + 2.47·28-s − 1.22·29-s + 0.0526·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3267550807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3267550807\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 5.36T + 8T^{2} \) |
| 3 | \( 1 + 0.836T + 27T^{2} \) |
| 5 | \( 1 - 1.92T + 125T^{2} \) |
| 7 | \( 1 - 17.6T + 343T^{2} \) |
| 17 | \( 1 + 29.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 78.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 167.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 191.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 47.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 27.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 297.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 522.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 27.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 179.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 92.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 824.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 224.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 461.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 883.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 150.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 328.T + 9.12e5T^{2} \) |
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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
−8.699194408199666181501198557356, −8.224983637893502566100589542020, −7.76118901686021031241456101181, −6.67198691788033027625771155901, −6.02707465559685640385549573012, −5.09527406743936423588338099543, −3.58017830215710845572935781432, −2.18886290455289526968714077637, −1.83574296935831170657758754385, −0.34087341703211448006443074935,
0.34087341703211448006443074935, 1.83574296935831170657758754385, 2.18886290455289526968714077637, 3.58017830215710845572935781432, 5.09527406743936423588338099543, 6.02707465559685640385549573012, 6.67198691788033027625771155901, 7.76118901686021031241456101181, 8.224983637893502566100589542020, 8.699194408199666181501198557356
