| L(s) = 1 | − 2·2-s − 28·4-s − 25·5-s − 168·7-s + 120·8-s + 50·10-s − 52·11-s + 169·13-s + 336·14-s + 656·16-s − 1.32e3·17-s + 1.70e3·19-s + 700·20-s + 104·22-s − 1.85e3·23-s + 625·25-s − 338·26-s + 4.70e3·28-s + 4.25e3·29-s + 7.19e3·31-s − 5.15e3·32-s + 2.64e3·34-s + 4.20e3·35-s − 2.29e3·37-s − 3.40e3·38-s − 3.00e3·40-s + 6.43e3·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 7/8·4-s − 0.447·5-s − 1.29·7-s + 0.662·8-s + 0.158·10-s − 0.129·11-s + 0.277·13-s + 0.458·14-s + 0.640·16-s − 1.10·17-s + 1.08·19-s + 0.391·20-s + 0.0458·22-s − 0.731·23-s + 1/5·25-s − 0.0980·26-s + 1.13·28-s + 0.938·29-s + 1.34·31-s − 0.889·32-s + 0.392·34-s + 0.579·35-s − 0.275·37-s − 0.381·38-s − 0.296·40-s + 0.598·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p^{2} T \) |
| 13 | \( 1 - p^{2} T \) |
| good | 2 | \( 1 + p T + p^{5} T^{2} \) |
| 7 | \( 1 + 24 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 52 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1322 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1700 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1856 T + p^{5} T^{2} \) |
| 29 | \( 1 - 4250 T + p^{5} T^{2} \) |
| 31 | \( 1 - 232 p T + p^{5} T^{2} \) |
| 37 | \( 1 + 2298 T + p^{5} T^{2} \) |
| 41 | \( 1 - 6438 T + p^{5} T^{2} \) |
| 43 | \( 1 - 18956 T + p^{5} T^{2} \) |
| 47 | \( 1 - 968 T + p^{5} T^{2} \) |
| 53 | \( 1 + 15366 T + p^{5} T^{2} \) |
| 59 | \( 1 - 2940 T + p^{5} T^{2} \) |
| 61 | \( 1 - 26542 T + p^{5} T^{2} \) |
| 67 | \( 1 + 43588 T + p^{5} T^{2} \) |
| 71 | \( 1 - 20688 T + p^{5} T^{2} \) |
| 73 | \( 1 - 24786 T + p^{5} T^{2} \) |
| 79 | \( 1 - 51760 T + p^{5} T^{2} \) |
| 83 | \( 1 + 31436 T + p^{5} T^{2} \) |
| 89 | \( 1 + 115690 T + p^{5} T^{2} \) |
| 97 | \( 1 + 127638 T + p^{5} T^{2} \) |
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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
−9.506673226390532770332751618181, −8.696963472165771109372160006641, −7.86545331380126429858986281615, −6.83745155263229154568555029610, −5.89087929085790369823351910794, −4.65320660213905323151087318995, −3.80389866012128326586126427219, −2.74248074749843168626481495601, −0.944175572299128921985124368349, 0,
0.944175572299128921985124368349, 2.74248074749843168626481495601, 3.80389866012128326586126427219, 4.65320660213905323151087318995, 5.89087929085790369823351910794, 6.83745155263229154568555029610, 7.86545331380126429858986281615, 8.696963472165771109372160006641, 9.506673226390532770332751618181
