| L(s) = 1 | − 1.51·3-s + 3.49·7-s − 0.705·9-s − 4.35·11-s − 3.67·13-s + 5.18·17-s + 2.08·19-s − 5.29·21-s − 23-s + 5.61·27-s − 1.24·29-s + 4.82·31-s + 6.59·33-s − 1.04·37-s + 5.57·39-s + 9.05·41-s − 10.4·43-s − 12.8·47-s + 5.24·49-s − 7.85·51-s − 9.37·53-s − 3.15·57-s + 14.1·59-s − 9.29·61-s − 2.46·63-s − 4.44·67-s + 1.51·69-s + ⋯ |
| L(s) = 1 | − 0.874·3-s + 1.32·7-s − 0.235·9-s − 1.31·11-s − 1.02·13-s + 1.25·17-s + 0.478·19-s − 1.15·21-s − 0.208·23-s + 1.08·27-s − 0.231·29-s + 0.866·31-s + 1.14·33-s − 0.171·37-s + 0.892·39-s + 1.41·41-s − 1.60·43-s − 1.87·47-s + 0.748·49-s − 1.09·51-s − 1.28·53-s − 0.418·57-s + 1.84·59-s − 1.19·61-s − 0.311·63-s − 0.542·67-s + 0.182·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 1.51T + 3T^{2} \) |
| 7 | \( 1 - 3.49T + 7T^{2} \) |
| 11 | \( 1 + 4.35T + 11T^{2} \) |
| 13 | \( 1 + 3.67T + 13T^{2} \) |
| 17 | \( 1 - 5.18T + 17T^{2} \) |
| 19 | \( 1 - 2.08T + 19T^{2} \) |
| 29 | \( 1 + 1.24T + 29T^{2} \) |
| 31 | \( 1 - 4.82T + 31T^{2} \) |
| 37 | \( 1 + 1.04T + 37T^{2} \) |
| 41 | \( 1 - 9.05T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + 9.37T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 + 9.29T + 61T^{2} \) |
| 67 | \( 1 + 4.44T + 67T^{2} \) |
| 71 | \( 1 - 2.45T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 7.72T + 79T^{2} \) |
| 83 | \( 1 - 2.97T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 9.37T + 97T^{2} \) |
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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
−8.076095136951339655116744993306, −7.35567119327785571320391321357, −6.41183420975522593440368068226, −5.44329461307201075738243405460, −5.15940940147053003680079585787, −4.59846637137409192373723603804, −3.24235432881110504975201712335, −2.39409358524364160843291578051, −1.25684170420085087120493260866, 0,
1.25684170420085087120493260866, 2.39409358524364160843291578051, 3.24235432881110504975201712335, 4.59846637137409192373723603804, 5.15940940147053003680079585787, 5.44329461307201075738243405460, 6.41183420975522593440368068226, 7.35567119327785571320391321357, 8.076095136951339655116744993306
