| L(s) = 1 | − 9.61·3-s − 16.7·5-s + 19.9·7-s + 65.3·9-s − 39.6·13-s + 160.·15-s + 2.06·17-s − 101.·19-s − 191.·21-s − 110.·23-s + 154.·25-s − 368.·27-s + 33.3·29-s − 173.·31-s − 332.·35-s + 251.·37-s + 380.·39-s + 192.·41-s + 304.·43-s − 1.09e3·45-s + 34.1·47-s + 53.7·49-s − 19.8·51-s + 216.·53-s + 975.·57-s + 16.0·59-s + 531.·61-s + ⋯ |
| L(s) = 1 | − 1.84·3-s − 1.49·5-s + 1.07·7-s + 2.42·9-s − 0.845·13-s + 2.76·15-s + 0.0293·17-s − 1.22·19-s − 1.98·21-s − 1.00·23-s + 1.23·25-s − 2.62·27-s + 0.213·29-s − 1.00·31-s − 1.60·35-s + 1.11·37-s + 1.56·39-s + 0.734·41-s + 1.07·43-s − 3.61·45-s + 0.105·47-s + 0.156·49-s − 0.0543·51-s + 0.561·53-s + 2.26·57-s + 0.0354·59-s + 1.11·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1936 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + 9.61T + 27T^{2} \) |
| 5 | \( 1 + 16.7T + 125T^{2} \) |
| 7 | \( 1 - 19.9T + 343T^{2} \) |
| 13 | \( 1 + 39.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 2.06T + 4.91e3T^{2} \) |
| 19 | \( 1 + 101.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 33.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 173.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 251.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 192.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 304.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 34.1T + 1.03e5T^{2} \) |
| 53 | \( 1 - 216.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 16.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 531.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 636.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 280.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 656.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 806.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 638.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 840.T + 9.12e5T^{2} \) |
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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
−8.067569551537981392385778161246, −7.64707919891833603338336633407, −6.86926669349856804701283261694, −5.98177174247741239849210248155, −5.12320437216622350930246965612, −4.40461120048414332988957884280, −3.98001906130175330898569999706, −2.11826947541270645882358240527, −0.813703754371671616636961479954, 0,
0.813703754371671616636961479954, 2.11826947541270645882358240527, 3.98001906130175330898569999706, 4.40461120048414332988957884280, 5.12320437216622350930246965612, 5.98177174247741239849210248155, 6.86926669349856804701283261694, 7.64707919891833603338336633407, 8.067569551537981392385778161246
