L(s) = 1 | + 2-s − 3-s + 4-s − 1.28·5-s − 6-s − 0.150·7-s + 8-s + 9-s − 1.28·10-s + 2.72·11-s − 12-s − 13-s − 0.150·14-s + 1.28·15-s + 16-s − 1.40·17-s + 18-s + 2.98·19-s − 1.28·20-s + 0.150·21-s + 2.72·22-s − 3.64·23-s − 24-s − 3.35·25-s − 26-s − 27-s − 0.150·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.572·5-s − 0.408·6-s − 0.0570·7-s + 0.353·8-s + 0.333·9-s − 0.405·10-s + 0.822·11-s − 0.288·12-s − 0.277·13-s − 0.0403·14-s + 0.330·15-s + 0.250·16-s − 0.339·17-s + 0.235·18-s + 0.684·19-s − 0.286·20-s + 0.0329·21-s + 0.581·22-s − 0.759·23-s − 0.204·24-s − 0.671·25-s − 0.196·26-s − 0.192·27-s − 0.0285·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 - T \) |
good | 5 | \( 1 + 1.28T + 5T^{2} \) |
| 7 | \( 1 + 0.150T + 7T^{2} \) |
| 11 | \( 1 - 2.72T + 11T^{2} \) |
| 17 | \( 1 + 1.40T + 17T^{2} \) |
| 19 | \( 1 - 2.98T + 19T^{2} \) |
| 23 | \( 1 + 3.64T + 23T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 - 5.90T + 37T^{2} \) |
| 41 | \( 1 + 3.58T + 41T^{2} \) |
| 43 | \( 1 - 4.00T + 43T^{2} \) |
| 47 | \( 1 + 7.94T + 47T^{2} \) |
| 53 | \( 1 - 9.76T + 53T^{2} \) |
| 59 | \( 1 - 5.94T + 59T^{2} \) |
| 61 | \( 1 + 0.902T + 61T^{2} \) |
| 67 | \( 1 + 9.25T + 67T^{2} \) |
| 71 | \( 1 + 1.84T + 71T^{2} \) |
| 73 | \( 1 - 0.335T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 - 16.7T + 83T^{2} \) |
| 89 | \( 1 + 2.76T + 89T^{2} \) |
| 97 | \( 1 - 4.08T + 97T^{2} \) |
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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
−7.39626538095462112825031086186, −6.68047457206199743357903299786, −6.04041916898065599977443581310, −5.41289216316158658378227856151, −4.59289592551817300248900615862, −3.97945763316860936326472623979, −3.38817396242421452687222582671, −2.27109549342622252103139416466, −1.31527434709106754142443480091, 0,
1.31527434709106754142443480091, 2.27109549342622252103139416466, 3.38817396242421452687222582671, 3.97945763316860936326472623979, 4.59289592551817300248900615862, 5.41289216316158658378227856151, 6.04041916898065599977443581310, 6.68047457206199743357903299786, 7.39626538095462112825031086186