L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 7-s + 8-s − 2·9-s − 2·11-s + 12-s + 5·13-s − 14-s + 16-s + 17-s − 2·18-s + 19-s − 21-s − 2·22-s + 24-s + 5·26-s − 5·27-s − 28-s + 29-s + 7·31-s + 32-s − 2·33-s + 34-s − 2·36-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s + 0.353·8-s − 2/3·9-s − 0.603·11-s + 0.288·12-s + 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.242·17-s − 0.471·18-s + 0.229·19-s − 0.218·21-s − 0.426·22-s + 0.204·24-s + 0.980·26-s − 0.962·27-s − 0.188·28-s + 0.185·29-s + 1.25·31-s + 0.176·32-s − 0.348·33-s + 0.171·34-s − 1/3·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.730269979\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.730269979\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 11 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 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
−8.140922340959310669082133871071, −7.38134792177791876506863078654, −6.53088356612856006482577107497, −5.85642580799879897006215029829, −5.34689414079455564901707452060, −4.28313714872669653698313194294, −3.57151068774675910315581389349, −2.92763195313147598726882829022, −2.20058156205665599755808941512, −0.891827231762930752730834699185,
0.891827231762930752730834699185, 2.20058156205665599755808941512, 2.92763195313147598726882829022, 3.57151068774675910315581389349, 4.28313714872669653698313194294, 5.34689414079455564901707452060, 5.85642580799879897006215029829, 6.53088356612856006482577107497, 7.38134792177791876506863078654, 8.140922340959310669082133871071