| L(s) = 1 | − 3·3-s − 7-s + 6·9-s − 5·11-s − 4·17-s + 2·19-s + 3·21-s + 5·23-s − 5·25-s − 9·27-s − 4·29-s − 31-s + 15·33-s + 7·37-s + 9·41-s + 12·43-s + 7·47-s + 49-s + 12·51-s + 4·53-s − 6·57-s − 6·59-s − 13·61-s − 6·63-s + 11·67-s − 15·69-s − 7·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2·9-s − 1.50·11-s − 0.970·17-s + 0.458·19-s + 0.654·21-s + 1.04·23-s − 25-s − 1.73·27-s − 0.742·29-s − 0.179·31-s + 2.61·33-s + 1.15·37-s + 1.40·41-s + 1.82·43-s + 1.02·47-s + 1/7·49-s + 1.68·51-s + 0.549·53-s − 0.794·57-s − 0.781·59-s − 1.66·61-s − 0.755·63-s + 1.34·67-s − 1.80·69-s − 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 17 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−14.21005833038994, −13.58186344167981, −13.10029197006136, −12.75568550822831, −12.39604898642745, −11.71592363643303, −11.17869401136357, −10.84681999136268, −10.61096871813636, −9.827065215107302, −9.445746014012855, −8.878700611445276, −7.973614067131046, −7.353079246128614, −7.255193076644114, −6.375711532655310, −5.819740347751253, −5.631150716121131, −5.008561315255179, −4.333434829095642, −4.018276814882257, −2.835766436655319, −2.463275648323300, −1.430361004424261, −0.6371668902493545, 0,
0.6371668902493545, 1.430361004424261, 2.463275648323300, 2.835766436655319, 4.018276814882257, 4.333434829095642, 5.008561315255179, 5.631150716121131, 5.819740347751253, 6.375711532655310, 7.255193076644114, 7.353079246128614, 7.973614067131046, 8.878700611445276, 9.445746014012855, 9.827065215107302, 10.61096871813636, 10.84681999136268, 11.17869401136357, 11.71592363643303, 12.39604898642745, 12.75568550822831, 13.10029197006136, 13.58186344167981, 14.21005833038994
