| L(s) = 1 | + 3·3-s + 13.0·5-s − 31.0·7-s + 9·9-s − 59.5·11-s + 62.8·13-s + 39.0·15-s − 97.4·17-s + 19·19-s − 93.0·21-s + 160.·23-s + 44.7·25-s + 27·27-s + 300.·29-s + 62.6·31-s − 178.·33-s − 403.·35-s + 91.7·37-s + 188.·39-s + 394.·41-s + 395.·43-s + 117.·45-s + 515.·47-s + 618.·49-s − 292.·51-s − 402.·53-s − 775.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.16·5-s − 1.67·7-s + 0.333·9-s − 1.63·11-s + 1.34·13-s + 0.672·15-s − 1.39·17-s + 0.229·19-s − 0.966·21-s + 1.45·23-s + 0.357·25-s + 0.192·27-s + 1.92·29-s + 0.363·31-s − 0.941·33-s − 1.95·35-s + 0.407·37-s + 0.774·39-s + 1.50·41-s + 1.40·43-s + 0.388·45-s + 1.59·47-s + 1.80·49-s − 0.802·51-s − 1.04·53-s − 1.90·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.525387258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.525387258\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 - 13.0T + 125T^{2} \) |
| 7 | \( 1 + 31.0T + 343T^{2} \) |
| 11 | \( 1 + 59.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 62.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 97.4T + 4.91e3T^{2} \) |
| 23 | \( 1 - 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 300.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 62.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.7T + 5.06e4T^{2} \) |
| 41 | \( 1 - 394.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 395.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 515.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 402.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 229.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 16.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 32.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 141.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 667.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 510.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 201.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 346.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 490.T + 9.12e5T^{2} \) |
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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
−9.538983291054276484092685449537, −9.096634968201919250848416551504, −8.192709556915279090111062251500, −6.96855303256963707726643991923, −6.29058909551605729342865065264, −5.54174249505081129608926673380, −4.25536218624779739203828570705, −2.87917887888725941512593686952, −2.54418478637511056408291297518, −0.830630848093114773080920801294,
0.830630848093114773080920801294, 2.54418478637511056408291297518, 2.87917887888725941512593686952, 4.25536218624779739203828570705, 5.54174249505081129608926673380, 6.29058909551605729342865065264, 6.96855303256963707726643991923, 8.192709556915279090111062251500, 9.096634968201919250848416551504, 9.538983291054276484092685449537
