| L(s) = 1 | + 8.45·3-s + 14.4·5-s − 27.3·7-s + 44.5·9-s + 58·11-s + 13·13-s + 122.·15-s − 37.2·17-s + 28.1·19-s − 231.·21-s + 205.·23-s + 84.0·25-s + 148.·27-s + 6·29-s − 202.·31-s + 490.·33-s − 395.·35-s + 187.·37-s + 109.·39-s − 336.·41-s − 204.·43-s + 643.·45-s + 370.·47-s + 406.·49-s − 314.·51-s − 64.2·53-s + 838.·55-s + ⋯ |
| L(s) = 1 | + 1.62·3-s + 1.29·5-s − 1.47·7-s + 1.64·9-s + 1.58·11-s + 0.277·13-s + 2.10·15-s − 0.530·17-s + 0.340·19-s − 2.40·21-s + 1.86·23-s + 0.672·25-s + 1.05·27-s + 0.0384·29-s − 1.17·31-s + 2.58·33-s − 1.91·35-s + 0.835·37-s + 0.451·39-s − 1.28·41-s − 0.726·43-s + 2.13·45-s + 1.15·47-s + 1.18·49-s − 0.864·51-s − 0.166·53-s + 2.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.710432215\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.710432215\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 - 8.45T + 27T^{2} \) |
| 5 | \( 1 - 14.4T + 125T^{2} \) |
| 7 | \( 1 + 27.3T + 343T^{2} \) |
| 11 | \( 1 - 58T + 1.33e3T^{2} \) |
| 17 | \( 1 + 37.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 28.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 205.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 202.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 187.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 336.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 204.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 370.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 64.2T + 1.48e5T^{2} \) |
| 59 | \( 1 - 784.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 815.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 40.4T + 3.57e5T^{2} \) |
| 73 | \( 1 + 104.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 82.6T + 4.93e5T^{2} \) |
| 83 | \( 1 + 833.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 795.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 188.T + 9.12e5T^{2} \) |
| 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
−9.471842830991494465474645041024, −9.195905225845445322179757745507, −8.529040033471456146111839533862, −6.95378905136179733736024568391, −6.71183191772306024389692947003, −5.53094170918388284789996120635, −3.96419340480302127615832325086, −3.24033605899437219108077351411, −2.33530728722036568084927128298, −1.24122346578891067513414252201,
1.24122346578891067513414252201, 2.33530728722036568084927128298, 3.24033605899437219108077351411, 3.96419340480302127615832325086, 5.53094170918388284789996120635, 6.71183191772306024389692947003, 6.95378905136179733736024568391, 8.529040033471456146111839533862, 9.195905225845445322179757745507, 9.471842830991494465474645041024
