| L(s) = 1 | + 8·5-s − 12·7-s + 12·11-s + 20·13-s − 62·17-s + 108·19-s − 72·23-s − 61·25-s + 128·29-s − 204·31-s − 96·35-s − 228·37-s − 22·41-s − 204·43-s + 600·47-s − 199·49-s − 256·53-s + 96·55-s + 828·59-s − 84·61-s + 160·65-s + 348·67-s + 456·71-s − 822·73-s − 144·77-s − 1.35e3·79-s − 108·83-s + ⋯ |
| L(s) = 1 | + 0.715·5-s − 0.647·7-s + 0.328·11-s + 0.426·13-s − 0.884·17-s + 1.30·19-s − 0.652·23-s − 0.487·25-s + 0.819·29-s − 1.18·31-s − 0.463·35-s − 1.01·37-s − 0.0838·41-s − 0.723·43-s + 1.86·47-s − 0.580·49-s − 0.663·53-s + 0.235·55-s + 1.82·59-s − 0.176·61-s + 0.305·65-s + 0.634·67-s + 0.762·71-s − 1.31·73-s − 0.213·77-s − 1.93·79-s − 0.142·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 8 T + p^{3} T^{2} \) |
| 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 62 T + p^{3} T^{2} \) |
| 19 | \( 1 - 108 T + p^{3} T^{2} \) |
| 23 | \( 1 + 72 T + p^{3} T^{2} \) |
| 29 | \( 1 - 128 T + p^{3} T^{2} \) |
| 31 | \( 1 + 204 T + p^{3} T^{2} \) |
| 37 | \( 1 + 228 T + p^{3} T^{2} \) |
| 41 | \( 1 + 22 T + p^{3} T^{2} \) |
| 43 | \( 1 + 204 T + p^{3} T^{2} \) |
| 47 | \( 1 - 600 T + p^{3} T^{2} \) |
| 53 | \( 1 + 256 T + p^{3} T^{2} \) |
| 59 | \( 1 - 828 T + p^{3} T^{2} \) |
| 61 | \( 1 + 84 T + p^{3} T^{2} \) |
| 67 | \( 1 - 348 T + p^{3} T^{2} \) |
| 71 | \( 1 - 456 T + p^{3} T^{2} \) |
| 73 | \( 1 + 822 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1356 T + p^{3} T^{2} \) |
| 83 | \( 1 + 108 T + p^{3} T^{2} \) |
| 89 | \( 1 + 938 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1278 T + p^{3} T^{2} \) |
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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
−8.437469214845024449943320130435, −7.34440359400884216030968739258, −6.69181514486943130346927047947, −5.90965621464589364587696812415, −5.27899426665296927346009267346, −4.12170249098742425229030902099, −3.31797267153417788832293193865, −2.27950807547226978148831902472, −1.31093180654452234775958741402, 0,
1.31093180654452234775958741402, 2.27950807547226978148831902472, 3.31797267153417788832293193865, 4.12170249098742425229030902099, 5.27899426665296927346009267346, 5.90965621464589364587696812415, 6.69181514486943130346927047947, 7.34440359400884216030968739258, 8.437469214845024449943320130435
