| L(s) = 1 | − 7.68·3-s + 15.2·5-s − 2.79·7-s + 32.0·9-s + 0.456·11-s + 0.618·13-s − 117.·15-s + 12.2·17-s + 19·19-s + 21.4·21-s − 116.·23-s + 107.·25-s − 38.9·27-s − 100.·29-s − 206.·31-s − 3.50·33-s − 42.6·35-s + 212.·37-s − 4.75·39-s − 42.1·41-s + 34.2·43-s + 489.·45-s − 393.·47-s − 335.·49-s − 94.1·51-s + 497.·53-s + 6.96·55-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + 1.36·5-s − 0.150·7-s + 1.18·9-s + 0.0125·11-s + 0.0131·13-s − 2.01·15-s + 0.174·17-s + 0.229·19-s + 0.223·21-s − 1.05·23-s + 0.863·25-s − 0.277·27-s − 0.644·29-s − 1.19·31-s − 0.0184·33-s − 0.205·35-s + 0.945·37-s − 0.0195·39-s − 0.160·41-s + 0.121·43-s + 1.62·45-s − 1.22·47-s − 0.977·49-s − 0.258·51-s + 1.28·53-s + 0.0170·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 3 | \( 1 + 7.68T + 27T^{2} \) |
| 5 | \( 1 - 15.2T + 125T^{2} \) |
| 7 | \( 1 + 2.79T + 343T^{2} \) |
| 11 | \( 1 - 0.456T + 1.33e3T^{2} \) |
| 13 | \( 1 - 0.618T + 2.19e3T^{2} \) |
| 17 | \( 1 - 12.2T + 4.91e3T^{2} \) |
| 23 | \( 1 + 116.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 100.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 212.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 42.1T + 6.89e4T^{2} \) |
| 43 | \( 1 - 34.2T + 7.95e4T^{2} \) |
| 47 | \( 1 + 393.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 497.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 82.9T + 2.05e5T^{2} \) |
| 61 | \( 1 - 655.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 88.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 430.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 65.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 528.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 592.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 51.7T + 9.12e5T^{2} \) |
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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
−9.319437399019489429143972445514, −8.066690277842389380064089142647, −6.92809052146984393245344588560, −6.24717851582589412129467460039, −5.60497434672421806321038451602, −5.07978191360034121922231946162, −3.82857280868457835305747100445, −2.30595723567579981449013304270, −1.27908977338653339780109121142, 0,
1.27908977338653339780109121142, 2.30595723567579981449013304270, 3.82857280868457835305747100445, 5.07978191360034121922231946162, 5.60497434672421806321038451602, 6.24717851582589412129467460039, 6.92809052146984393245344588560, 8.066690277842389380064089142647, 9.319437399019489429143972445514
