| L(s) = 1 | + 2-s + 4-s + 8-s + 4.24·11-s + 6.24·13-s + 16-s + 6.24·19-s + 4.24·22-s + 7.24·23-s − 5·25-s + 6.24·26-s − 4.24·29-s + 0.757·31-s + 32-s − 4·37-s + 6.24·38-s − 5.48·41-s − 6.48·43-s + 4.24·44-s + 7.24·46-s + 13.2·47-s − 5·50-s + 6.24·52-s + 4.24·53-s − 4.24·58-s + 6.24·61-s + 0.757·62-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.353·8-s + 1.27·11-s + 1.73·13-s + 0.250·16-s + 1.43·19-s + 0.904·22-s + 1.51·23-s − 25-s + 1.22·26-s − 0.787·29-s + 0.136·31-s + 0.176·32-s − 0.657·37-s + 1.01·38-s − 0.856·41-s − 0.988·43-s + 0.639·44-s + 1.06·46-s + 1.93·47-s − 0.707·50-s + 0.865·52-s + 0.582·53-s − 0.557·58-s + 0.799·61-s + 0.0961·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.411853647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.411853647\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 4.24T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 6.24T + 19T^{2} \) |
| 23 | \( 1 - 7.24T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 0.757T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 5.48T + 41T^{2} \) |
| 43 | \( 1 + 6.48T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 - 4.24T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 6.24T + 61T^{2} \) |
| 67 | \( 1 + 8.24T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 + 7T + 73T^{2} \) |
| 79 | \( 1 - 9.24T + 79T^{2} \) |
| 83 | \( 1 - 7.75T + 83T^{2} \) |
| 89 | \( 1 + 5.48T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−7.64881818274005793129962247361, −7.00771464529860782769396660494, −6.41217437750548102955904505747, −5.69018829035559722487623337642, −5.16219504112330216213752204497, −4.05906611938360777203433040467, −3.66723363976203350272019754494, −2.96386756593487151351751450024, −1.66509796729960218976711994202, −1.05337508707568473600596514998,
1.05337508707568473600596514998, 1.66509796729960218976711994202, 2.96386756593487151351751450024, 3.66723363976203350272019754494, 4.05906611938360777203433040467, 5.16219504112330216213752204497, 5.69018829035559722487623337642, 6.41217437750548102955904505747, 7.00771464529860782769396660494, 7.64881818274005793129962247361
