| L(s) = 1 | + 3-s − 0.585·5-s + 2.82·7-s + 9-s − 2·11-s + 2.82·13-s − 0.585·15-s − 2.58·17-s − 6.82·19-s + 2.82·21-s + 5.65·23-s − 4.65·25-s + 27-s + 2·29-s − 1.17·31-s − 2·33-s − 1.65·35-s − 4·37-s + 2.82·39-s − 2.58·41-s + 8.24·43-s − 0.585·45-s + 7.17·47-s + 1.00·49-s − 2.58·51-s + 2.24·53-s + 1.17·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.261·5-s + 1.06·7-s + 0.333·9-s − 0.603·11-s + 0.784·13-s − 0.151·15-s − 0.627·17-s − 1.56·19-s + 0.617·21-s + 1.17·23-s − 0.931·25-s + 0.192·27-s + 0.371·29-s − 0.210·31-s − 0.348·33-s − 0.280·35-s − 0.657·37-s + 0.452·39-s − 0.403·41-s + 1.25·43-s − 0.0873·45-s + 1.04·47-s + 0.142·49-s − 0.362·51-s + 0.308·53-s + 0.157·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.677069872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.677069872\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 167 | \( 1 - T \) |
| good | 5 | \( 1 + 0.585T + 5T^{2} \) |
| 7 | \( 1 - 2.82T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 2.82T + 13T^{2} \) |
| 17 | \( 1 + 2.58T + 17T^{2} \) |
| 19 | \( 1 + 6.82T + 19T^{2} \) |
| 23 | \( 1 - 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 2.58T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 - 7.17T + 47T^{2} \) |
| 53 | \( 1 - 2.24T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 + 1.41T + 67T^{2} \) |
| 71 | \( 1 - 12.4T + 71T^{2} \) |
| 73 | \( 1 - 12.1T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 11.6T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + 17.3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.180103428393529296206690289712, −7.15995143270332002919940674372, −6.64618330549798191451695758216, −5.64132984961816535481468629611, −4.97549909455134893697845209193, −4.17667630494894134572701322074, −3.66860569822804063758825855589, −2.46718406640213291649726689426, −1.98034524652463485856812744997, −0.789276526356961023768938275884,
0.789276526356961023768938275884, 1.98034524652463485856812744997, 2.46718406640213291649726689426, 3.66860569822804063758825855589, 4.17667630494894134572701322074, 4.97549909455134893697845209193, 5.64132984961816535481468629611, 6.64618330549798191451695758216, 7.15995143270332002919940674372, 8.180103428393529296206690289712
