| L(s) = 1 | − 3.08·2-s + 1.53·4-s + 31.3·7-s + 19.9·8-s + 19.1·11-s + 20.9·13-s − 96.6·14-s − 73.9·16-s − 6.19·17-s − 96.6·19-s − 59.2·22-s + 162.·23-s − 64.8·26-s + 47.9·28-s − 7.64·29-s + 225.·31-s + 68.4·32-s + 19.1·34-s − 155.·37-s + 298.·38-s + 315.·41-s + 192.·43-s + 29.4·44-s − 503.·46-s + 318.·47-s + 637.·49-s + 32.1·52-s + ⋯ |
| L(s) = 1 | − 1.09·2-s + 0.191·4-s + 1.69·7-s + 0.882·8-s + 0.526·11-s + 0.447·13-s − 1.84·14-s − 1.15·16-s − 0.0883·17-s − 1.16·19-s − 0.574·22-s + 1.47·23-s − 0.488·26-s + 0.323·28-s − 0.0489·29-s + 1.30·31-s + 0.378·32-s + 0.0964·34-s − 0.692·37-s + 1.27·38-s + 1.20·41-s + 0.683·43-s + 0.100·44-s − 1.61·46-s + 0.987·47-s + 1.85·49-s + 0.0857·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2025 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.675594869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.675594869\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 3.08T + 8T^{2} \) |
| 7 | \( 1 - 31.3T + 343T^{2} \) |
| 11 | \( 1 - 19.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 20.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 6.19T + 4.91e3T^{2} \) |
| 19 | \( 1 + 96.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 162.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 7.64T + 2.43e4T^{2} \) |
| 31 | \( 1 - 225.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 155.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 315.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 318.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 277.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 429.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 89.9T + 2.26e5T^{2} \) |
| 67 | \( 1 + 583.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 132.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 259.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 124.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 36.7T + 5.71e5T^{2} \) |
| 89 | \( 1 + 333.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.02e3T + 9.12e5T^{2} \) |
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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
−8.739245706203375285755529657492, −8.244923166753885061176669910087, −7.48591689104209150525486926145, −6.73682155145461724353099661401, −5.57234404832519836187113527289, −4.61340137847169738116380440263, −4.11893535841174124590686337530, −2.47105703308799845234420562199, −1.47274203793044631473458190187, −0.795459719974522991118828433693,
0.795459719974522991118828433693, 1.47274203793044631473458190187, 2.47105703308799845234420562199, 4.11893535841174124590686337530, 4.61340137847169738116380440263, 5.57234404832519836187113527289, 6.73682155145461724353099661401, 7.48591689104209150525486926145, 8.244923166753885061176669910087, 8.739245706203375285755529657492
