| L(s) = 1 | − 5.26·2-s − 4.31·4-s − 131.·7-s + 191.·8-s − 290.·11-s − 68.3·13-s + 689.·14-s − 867.·16-s − 310.·17-s − 2.13e3·19-s + 1.52e3·22-s + 873.·23-s + 359.·26-s + 564.·28-s + 2.58e3·29-s − 9.08e3·31-s − 1.54e3·32-s + 1.63e3·34-s − 3.99e3·37-s + 1.12e4·38-s − 1.69e4·41-s + 1.80e4·43-s + 1.25e3·44-s − 4.59e3·46-s + 2.48e4·47-s + 366.·49-s + 294.·52-s + ⋯ |
| L(s) = 1 | − 0.930·2-s − 0.134·4-s − 1.01·7-s + 1.05·8-s − 0.722·11-s − 0.112·13-s + 0.940·14-s − 0.847·16-s − 0.260·17-s − 1.35·19-s + 0.672·22-s + 0.344·23-s + 0.104·26-s + 0.136·28-s + 0.569·29-s − 1.69·31-s − 0.267·32-s + 0.242·34-s − 0.479·37-s + 1.26·38-s − 1.57·41-s + 1.48·43-s + 0.0973·44-s − 0.320·46-s + 1.64·47-s + 0.0218·49-s + 0.0151·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4911601550\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4911601550\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 5.26T + 32T^{2} \) |
| 7 | \( 1 + 131.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 290.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 68.3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 310.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.13e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 873.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.58e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.08e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.99e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.69e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.80e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.48e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.65e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 9.23e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.32e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.23e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.58e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.65e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 7.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.96e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.17e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.18e4T + 8.58e9T^{2} \) |
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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
−10.90880775493321453296346416087, −10.32922925742657441238937397695, −9.338195420571021429742679796303, −8.622535718665849500450433477217, −7.53550758969185555618980251413, −6.51992904178314121938969783543, −5.11640618532413895898539994873, −3.77650973962669880183824624873, −2.19946919416405276868563605344, −0.46631085599505725640030348152,
0.46631085599505725640030348152, 2.19946919416405276868563605344, 3.77650973962669880183824624873, 5.11640618532413895898539994873, 6.51992904178314121938969783543, 7.53550758969185555618980251413, 8.622535718665849500450433477217, 9.338195420571021429742679796303, 10.32922925742657441238937397695, 10.90880775493321453296346416087
