| L(s) = 1 | + 3·3-s + 5-s + 7-s + 6·9-s + 5·11-s − 3·13-s + 3·15-s + 17-s − 6·19-s + 3·21-s + 25-s + 9·27-s − 9·29-s − 4·31-s + 15·33-s + 35-s − 2·37-s − 9·39-s − 4·41-s − 10·43-s + 6·45-s − 47-s + 49-s + 3·51-s − 4·53-s + 5·55-s − 18·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s + 1.50·11-s − 0.832·13-s + 0.774·15-s + 0.242·17-s − 1.37·19-s + 0.654·21-s + 1/5·25-s + 1.73·27-s − 1.67·29-s − 0.718·31-s + 2.61·33-s + 0.169·35-s − 0.328·37-s − 1.44·39-s − 0.624·41-s − 1.52·43-s + 0.894·45-s − 0.145·47-s + 1/7·49-s + 0.420·51-s − 0.549·53-s + 0.674·55-s − 2.38·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74060 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74060 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−14.46309943154321, −14.01597557802883, −13.33062881786697, −13.04819802163714, −12.48870794063782, −11.91978912076470, −11.35777643751650, −10.68621987097805, −10.10162791493015, −9.602822853205946, −9.210456634257762, −8.795407064647561, −8.345351710605215, −7.747088918595548, −7.221124325699604, −6.740983363943207, −6.187048169434523, −5.348123731876551, −4.699801349266266, −4.061330310365020, −3.656797909575803, −3.057887375663727, −2.267271328144395, −1.767455766455058, −1.436455408749587, 0,
1.436455408749587, 1.767455766455058, 2.267271328144395, 3.057887375663727, 3.656797909575803, 4.061330310365020, 4.699801349266266, 5.348123731876551, 6.187048169434523, 6.740983363943207, 7.221124325699604, 7.747088918595548, 8.345351710605215, 8.795407064647561, 9.210456634257762, 9.602822853205946, 10.10162791493015, 10.68621987097805, 11.35777643751650, 11.91978912076470, 12.48870794063782, 13.04819802163714, 13.33062881786697, 14.01597557802883, 14.46309943154321
