| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 2·11-s + 12-s − 4·13-s + 16-s − 5·17-s + 18-s + 4·19-s − 2·22-s − 5·23-s + 24-s − 4·26-s + 27-s − 6·29-s + 11·31-s + 32-s − 2·33-s − 5·34-s + 36-s − 8·37-s + 4·38-s − 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.10·13-s + 1/4·16-s − 1.21·17-s + 0.235·18-s + 0.917·19-s − 0.426·22-s − 1.04·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s − 1.11·29-s + 1.97·31-s + 0.176·32-s − 0.348·33-s − 0.857·34-s + 1/6·36-s − 1.31·37-s + 0.648·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 11 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.60415391122940988606663623032, −6.82672208400176975629931603032, −6.21312793290483321344669364509, −5.20015272251918925694853062990, −4.77242315405532644472704103449, −3.95442891474874095245767771289, −3.09575730921477893769998690737, −2.43939617880774432059133899002, −1.66490464031237872533052392577, 0,
1.66490464031237872533052392577, 2.43939617880774432059133899002, 3.09575730921477893769998690737, 3.95442891474874095245767771289, 4.77242315405532644472704103449, 5.20015272251918925694853062990, 6.21312793290483321344669364509, 6.82672208400176975629931603032, 7.60415391122940988606663623032
