| L(s) = 1 | − 2-s − 3·3-s − 7·4-s − 5·5-s + 3·6-s + 33·7-s + 15·8-s + 9·9-s + 5·10-s + 21·12-s + 31·13-s − 33·14-s + 15·15-s + 41·16-s − 33·17-s − 9·18-s − 113·19-s + 35·20-s − 99·21-s − 44·23-s − 45·24-s + 25·25-s − 31·26-s − 27·27-s − 231·28-s + 51·29-s − 15·30-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 0.577·3-s − 7/8·4-s − 0.447·5-s + 0.204·6-s + 1.78·7-s + 0.662·8-s + 1/3·9-s + 0.158·10-s + 0.505·12-s + 0.661·13-s − 0.629·14-s + 0.258·15-s + 0.640·16-s − 0.470·17-s − 0.117·18-s − 1.36·19-s + 0.391·20-s − 1.02·21-s − 0.398·23-s − 0.382·24-s + 1/5·25-s − 0.233·26-s − 0.192·27-s − 1.55·28-s + 0.326·29-s − 0.0912·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 - 33 T + p^{3} T^{2} \) |
| 13 | \( 1 - 31 T + p^{3} T^{2} \) |
| 17 | \( 1 + 33 T + p^{3} T^{2} \) |
| 19 | \( 1 + 113 T + p^{3} T^{2} \) |
| 23 | \( 1 + 44 T + p^{3} T^{2} \) |
| 29 | \( 1 - 51 T + p^{3} T^{2} \) |
| 31 | \( 1 + 50 T + p^{3} T^{2} \) |
| 37 | \( 1 + 239 T + p^{3} T^{2} \) |
| 41 | \( 1 + 218 T + p^{3} T^{2} \) |
| 43 | \( 1 + 46 T + p^{3} T^{2} \) |
| 47 | \( 1 - 594 T + p^{3} T^{2} \) |
| 53 | \( 1 - 628 T + p^{3} T^{2} \) |
| 59 | \( 1 + 260 T + p^{3} T^{2} \) |
| 61 | \( 1 + 548 T + p^{3} T^{2} \) |
| 67 | \( 1 + 382 T + p^{3} T^{2} \) |
| 71 | \( 1 - 5 T + p^{3} T^{2} \) |
| 73 | \( 1 - 598 T + p^{3} T^{2} \) |
| 79 | \( 1 + 974 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1147 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1172 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1190 T + p^{3} T^{2} \) |
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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
−8.630937526153044262837221667382, −7.891985937205234272875279232020, −7.13343844155721476992012651258, −5.96502780826571387640621829669, −5.09308146857477316838446062922, −4.44279213768896593850455608924, −3.84979284500949007053845419945, −2.02806545122848318825355783733, −1.12011025227934270830883586965, 0,
1.12011025227934270830883586965, 2.02806545122848318825355783733, 3.84979284500949007053845419945, 4.44279213768896593850455608924, 5.09308146857477316838446062922, 5.96502780826571387640621829669, 7.13343844155721476992012651258, 7.891985937205234272875279232020, 8.630937526153044262837221667382
