| L(s) = 1 | + 0.910·2-s − 1.17·4-s − 5-s + 7-s − 2.88·8-s − 0.910·10-s − 6.14·11-s − 0.957·13-s + 0.910·14-s − 0.285·16-s − 7.38·17-s + 7.20·19-s + 1.17·20-s − 5.59·22-s + 23-s + 25-s − 0.872·26-s − 1.17·28-s + 5.63·29-s − 3.58·31-s + 5.51·32-s − 6.71·34-s − 35-s − 10.3·37-s + 6.56·38-s + 2.88·40-s − 2.09·41-s + ⋯ |
| L(s) = 1 | + 0.643·2-s − 0.585·4-s − 0.447·5-s + 0.377·7-s − 1.02·8-s − 0.287·10-s − 1.85·11-s − 0.265·13-s + 0.243·14-s − 0.0714·16-s − 1.79·17-s + 1.65·19-s + 0.261·20-s − 1.19·22-s + 0.208·23-s + 0.200·25-s − 0.171·26-s − 0.221·28-s + 1.04·29-s − 0.643·31-s + 0.974·32-s − 1.15·34-s − 0.169·35-s − 1.69·37-s + 1.06·38-s + 0.456·40-s − 0.327·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9680251207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9680251207\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - 0.910T + 2T^{2} \) |
| 11 | \( 1 + 6.14T + 11T^{2} \) |
| 13 | \( 1 + 0.957T + 13T^{2} \) |
| 17 | \( 1 + 7.38T + 17T^{2} \) |
| 19 | \( 1 - 7.20T + 19T^{2} \) |
| 29 | \( 1 - 5.63T + 29T^{2} \) |
| 31 | \( 1 + 3.58T + 31T^{2} \) |
| 37 | \( 1 + 10.3T + 37T^{2} \) |
| 41 | \( 1 + 2.09T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 + 12.5T + 47T^{2} \) |
| 53 | \( 1 - 0.521T + 53T^{2} \) |
| 59 | \( 1 - 6.80T + 59T^{2} \) |
| 61 | \( 1 - 7.27T + 61T^{2} \) |
| 67 | \( 1 + 1.03T + 67T^{2} \) |
| 71 | \( 1 - 2.96T + 71T^{2} \) |
| 73 | \( 1 + 5.64T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 - 11.9T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
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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
−8.043938942477474038288267841894, −7.17549490401804003805320855893, −6.54740824740181254325612778716, −5.35940259367925022118311809494, −5.08763463957792984224103431696, −4.59050216563331962155220822198, −3.51587582099790792076912417144, −2.97368499721848153044799881333, −1.99962195929515961368911515568, −0.43005956756673183494651018582,
0.43005956756673183494651018582, 1.99962195929515961368911515568, 2.97368499721848153044799881333, 3.51587582099790792076912417144, 4.59050216563331962155220822198, 5.08763463957792984224103431696, 5.35940259367925022118311809494, 6.54740824740181254325612778716, 7.17549490401804003805320855893, 8.043938942477474038288267841894
