| L(s) = 1 | − 3·3-s + 18.4·5-s + 22.4·7-s + 9·9-s + 53.6·11-s − 7.15·13-s − 55.2·15-s + 39.6·17-s + 125.·19-s − 67.2·21-s + 99.1·23-s + 214.·25-s − 27·27-s − 205.·29-s + 147.·31-s − 161.·33-s + 413.·35-s − 125.·37-s + 21.4·39-s − 506.·41-s − 413.·43-s + 165.·45-s − 313.·47-s + 159.·49-s − 119.·51-s + 44.3·53-s + 989.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.64·5-s + 1.21·7-s + 0.333·9-s + 1.47·11-s − 0.152·13-s − 0.951·15-s + 0.566·17-s + 1.51·19-s − 0.698·21-s + 0.898·23-s + 1.71·25-s − 0.192·27-s − 1.31·29-s + 0.854·31-s − 0.849·33-s + 1.99·35-s − 0.558·37-s + 0.0881·39-s − 1.92·41-s − 1.46·43-s + 0.549·45-s − 0.974·47-s + 0.465·49-s − 0.326·51-s + 0.114·53-s + 2.42·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.225810556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.225810556\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| good | 5 | \( 1 - 18.4T + 125T^{2} \) |
| 7 | \( 1 - 22.4T + 343T^{2} \) |
| 11 | \( 1 - 53.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 7.15T + 2.19e3T^{2} \) |
| 17 | \( 1 - 39.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 125.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 99.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 205.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 147.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 125.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 506.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 413.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 313.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 44.3T + 1.48e5T^{2} \) |
| 59 | \( 1 + 324T + 2.05e5T^{2} \) |
| 61 | \( 1 + 324T + 2.26e5T^{2} \) |
| 67 | \( 1 + 464.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.05e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 602.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 15.8T + 5.71e5T^{2} \) |
| 89 | \( 1 - 381.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 659.T + 9.12e5T^{2} \) |
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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
−9.779448909852847382546060161511, −9.376116940647493922457025597649, −8.303183989998309634106053644143, −7.09610897635939671834875333761, −6.35944992436599034656280145784, −5.33154424284670448438037634895, −4.93416257826124655930273353864, −3.37133580911363636842801467903, −1.73238364787003067350675067632, −1.25834291926636198745532918757,
1.25834291926636198745532918757, 1.73238364787003067350675067632, 3.37133580911363636842801467903, 4.93416257826124655930273353864, 5.33154424284670448438037634895, 6.35944992436599034656280145784, 7.09610897635939671834875333761, 8.303183989998309634106053644143, 9.376116940647493922457025597649, 9.779448909852847382546060161511
