| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 2·7-s + 8-s − 2·9-s + 12-s + 4·13-s − 2·14-s + 16-s + 3·17-s − 2·18-s − 5·19-s − 2·21-s + 6·23-s + 24-s + 4·26-s − 5·27-s − 2·28-s + 2·31-s + 32-s + 3·34-s − 2·36-s + 2·37-s − 5·38-s + 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.288·12-s + 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 1.14·19-s − 0.436·21-s + 1.25·23-s + 0.204·24-s + 0.784·26-s − 0.962·27-s − 0.377·28-s + 0.359·31-s + 0.176·32-s + 0.514·34-s − 1/3·36-s + 0.328·37-s − 0.811·38-s + 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.612648758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.612648758\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.003379760199122716602575955438, −7.37355750361837418705145948004, −6.39299024601606568879362228127, −6.05410683290230188214573869250, −5.25591445959428735491861815623, −4.25751151289549141315285174849, −3.54905277576412147109484999920, −2.96421704926700326951259240062, −2.18944587268591749591824641427, −0.870527168442070003633023384518,
0.870527168442070003633023384518, 2.18944587268591749591824641427, 2.96421704926700326951259240062, 3.54905277576412147109484999920, 4.25751151289549141315285174849, 5.25591445959428735491861815623, 6.05410683290230188214573869250, 6.39299024601606568879362228127, 7.37355750361837418705145948004, 8.003379760199122716602575955438
