| L(s) = 1 | − 2·2-s + 3·3-s + 4·4-s + 14.7·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s − 29.4·10-s + 42.3·11-s + 12·12-s + 10.0·13-s + 14·14-s + 44.2·15-s + 16·16-s + 130.·17-s − 18·18-s − 139.·19-s + 58.9·20-s − 21·21-s − 84.6·22-s + 23·23-s − 24·24-s + 92.5·25-s − 20.1·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.31·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.932·10-s + 1.16·11-s + 0.288·12-s + 0.214·13-s + 0.267·14-s + 0.761·15-s + 0.250·16-s + 1.86·17-s − 0.235·18-s − 1.68·19-s + 0.659·20-s − 0.218·21-s − 0.820·22-s + 0.208·23-s − 0.204·24-s + 0.740·25-s − 0.151·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.724050073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.724050073\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| 23 | \( 1 - 23T \) |
| good | 5 | \( 1 - 14.7T + 125T^{2} \) |
| 11 | \( 1 - 42.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 10.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 130.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 2.32T + 2.43e4T^{2} \) |
| 31 | \( 1 - 185.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 308.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 472.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 251.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 526.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 525.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 384.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 720.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 110.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 547.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 516.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 636.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 372.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 638.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 481.T + 9.12e5T^{2} \) |
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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
−9.699587961094695731758987092641, −8.891095905776524173924657052002, −8.238587087828115288327578903561, −7.07533447935865048165168164742, −6.31631617297475868675140055877, −5.62665560098082318997826440365, −4.11520369092260447895160903524, −2.98660476360558114865491952854, −1.93824135199440071808452703005, −1.03007736781869432437056785120,
1.03007736781869432437056785120, 1.93824135199440071808452703005, 2.98660476360558114865491952854, 4.11520369092260447895160903524, 5.62665560098082318997826440365, 6.31631617297475868675140055877, 7.07533447935865048165168164742, 8.238587087828115288327578903561, 8.891095905776524173924657052002, 9.699587961094695731758987092641
