L(s) = 1 | − 2-s + 3·3-s − 7·4-s + 5·5-s − 3·6-s + 24·7-s + 15·8-s + 9·9-s − 5·10-s − 21·12-s − 22·13-s − 24·14-s + 15·15-s + 41·16-s + 14·17-s − 9·18-s + 20·19-s − 35·20-s + 72·21-s − 168·23-s + 45·24-s + 25·25-s + 22·26-s + 27·27-s − 168·28-s − 230·29-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 + 1.29·7-s + 0.662·8-s + 1/3·9-s − 0.158·10-s − 0.505·12-s − 0.469·13-s − 0.458·14-s + 0.258·15-s + 0.640·16-s + 0.199·17-s − 0.117·18-s + 0.241·19-s − 0.391·20-s + 0.748·21-s − 1.52·23-s + 0.382·24-s + 1/5·25-s + 0.165·26-s + 0.192·27-s − 1.13·28-s − 1.47·29-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 | \( 1 - 24 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 230 T + p^{3} T^{2} \) |
| 31 | \( 1 + 288 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 + 122 T + p^{3} T^{2} \) |
| 43 | \( 1 - 188 T + p^{3} T^{2} \) |
| 47 | \( 1 - 256 T + p^{3} T^{2} \) |
| 53 | \( 1 + 338 T + p^{3} T^{2} \) |
| 59 | \( 1 - 100 T + p^{3} T^{2} \) |
| 61 | \( 1 + 742 T + p^{3} T^{2} \) |
| 67 | \( 1 + 84 T + p^{3} T^{2} \) |
| 71 | \( 1 + 328 T + p^{3} T^{2} \) |
| 73 | \( 1 - 38 T + p^{3} T^{2} \) |
| 79 | \( 1 - 240 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1212 T + p^{3} T^{2} \) |
| 89 | \( 1 - 330 T + p^{3} T^{2} \) |
| 97 | \( 1 - 866 T + p^{3} T^{2} \) |
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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
−8.641569477463814017183032857649, −7.64238058691923613510999831675, −7.53014566471801599835571048915, −5.90313607328616586839250961595, −5.18409271346909968904575429061, −4.37656358417063700981146737998, −3.55507148039555621237041862497, −2.09321481864337913153636490983, −1.45391834572287050326864920532, 0,
1.45391834572287050326864920532, 2.09321481864337913153636490983, 3.55507148039555621237041862497, 4.37656358417063700981146737998, 5.18409271346909968904575429061, 5.90313607328616586839250961595, 7.53014566471801599835571048915, 7.64238058691923613510999831675, 8.641569477463814017183032857649