| L(s) = 1 | + 3.32·3-s + 5-s + 2.39·7-s + 8.04·9-s − 11-s − 2·13-s + 3.32·15-s + 3.32·17-s + 6.24·19-s + 7.97·21-s + 2·23-s + 25-s + 16.7·27-s + 1.47·29-s − 7.97·31-s − 3.32·33-s + 2.39·35-s − 6.39·37-s − 6.64·39-s − 5.44·41-s − 12.0·43-s + 8.04·45-s − 5.44·47-s − 1.24·49-s + 11.0·51-s − 6.39·53-s − 55-s + ⋯ |
| L(s) = 1 | + 1.91·3-s + 0.447·5-s + 0.906·7-s + 2.68·9-s − 0.301·11-s − 0.554·13-s + 0.858·15-s + 0.806·17-s + 1.43·19-s + 1.73·21-s + 0.417·23-s + 0.200·25-s + 3.22·27-s + 0.273·29-s − 1.43·31-s − 0.578·33-s + 0.405·35-s − 1.05·37-s − 1.06·39-s − 0.850·41-s − 1.84·43-s + 1.19·45-s − 0.793·47-s − 0.178·49-s + 1.54·51-s − 0.878·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.910869470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.910869470\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 3.32T + 3T^{2} \) |
| 7 | \( 1 - 2.39T + 7T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 + 7.97T + 31T^{2} \) |
| 37 | \( 1 + 6.39T + 37T^{2} \) |
| 41 | \( 1 + 5.44T + 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 5.44T + 47T^{2} \) |
| 53 | \( 1 + 6.39T + 53T^{2} \) |
| 59 | \( 1 + 14.0T + 59T^{2} \) |
| 61 | \( 1 - 0.119T + 61T^{2} \) |
| 67 | \( 1 - 6.79T + 67T^{2} \) |
| 71 | \( 1 + 5.32T + 71T^{2} \) |
| 73 | \( 1 - 11.8T + 73T^{2} \) |
| 79 | \( 1 - 5.35T + 79T^{2} \) |
| 83 | \( 1 - 0.149T + 83T^{2} \) |
| 89 | \( 1 - 9.84T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−8.542982683662936316813772668046, −7.83273801446578043927530043236, −7.48130304391841623912273950826, −6.64342570725449577774100684539, −5.17100283447274733885428980500, −4.85721093788513368174288819929, −3.48044015008761845592412236635, −3.14718655998096588491028575283, −1.99330159827200088562508988077, −1.44310054328567771814257239482,
1.44310054328567771814257239482, 1.99330159827200088562508988077, 3.14718655998096588491028575283, 3.48044015008761845592412236635, 4.85721093788513368174288819929, 5.17100283447274733885428980500, 6.64342570725449577774100684539, 7.48130304391841623912273950826, 7.83273801446578043927530043236, 8.542982683662936316813772668046
