| L(s) = 1 | − 2-s + 0.00816·3-s + 4-s + 4.13·5-s − 0.00816·6-s − 5.05·7-s − 8-s − 2.99·9-s − 4.13·10-s + 1.63·11-s + 0.00816·12-s + 7.11·13-s + 5.05·14-s + 0.0337·15-s + 16-s − 4.74·17-s + 2.99·18-s − 2.86·19-s + 4.13·20-s − 0.0412·21-s − 1.63·22-s + 3.51·23-s − 0.00816·24-s + 12.0·25-s − 7.11·26-s − 0.0489·27-s − 5.05·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.00471·3-s + 0.5·4-s + 1.84·5-s − 0.00333·6-s − 1.91·7-s − 0.353·8-s − 0.999·9-s − 1.30·10-s + 0.493·11-s + 0.00235·12-s + 1.97·13-s + 1.35·14-s + 0.00870·15-s + 0.250·16-s − 1.15·17-s + 0.707·18-s − 0.656·19-s + 0.923·20-s − 0.00900·21-s − 0.348·22-s + 0.732·23-s − 0.00166·24-s + 2.41·25-s − 1.39·26-s − 0.00942·27-s − 0.955·28-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
| \(L(1)\) |
\(\approx\) |
\(1.577840920\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.577840920\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.00816T + 3T^{2} \) |
| 5 | \( 1 - 4.13T + 5T^{2} \) |
| 7 | \( 1 + 5.05T + 7T^{2} \) |
| 11 | \( 1 - 1.63T + 11T^{2} \) |
| 13 | \( 1 - 7.11T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 3.51T + 23T^{2} \) |
| 29 | \( 1 - 6.31T + 29T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 - 3.09T + 43T^{2} \) |
| 47 | \( 1 + 3.16T + 47T^{2} \) |
| 53 | \( 1 - 4.85T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 6.18T + 61T^{2} \) |
| 67 | \( 1 + 3.36T + 67T^{2} \) |
| 71 | \( 1 + 16.4T + 71T^{2} \) |
| 73 | \( 1 + 14.0T + 73T^{2} \) |
| 79 | \( 1 + 2.59T + 79T^{2} \) |
| 83 | \( 1 - 3.02T + 83T^{2} \) |
| 89 | \( 1 - 1.04T + 89T^{2} \) |
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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
−8.606974175153410102502577222505, −7.11117821432334432165153847816, −6.37632649795496665459590254035, −6.12356278892795467622777971093, −5.83239011824644135353385908347, −4.41128572230868561022435326536, −3.17821508930324419822166440523, −2.80365439620212615314478940305, −1.78940268176813611045872582511, −0.73160432789714082057572191836,
0.73160432789714082057572191836, 1.78940268176813611045872582511, 2.80365439620212615314478940305, 3.17821508930324419822166440523, 4.41128572230868561022435326536, 5.83239011824644135353385908347, 6.12356278892795467622777971093, 6.37632649795496665459590254035, 7.11117821432334432165153847816, 8.606974175153410102502577222505
