L(s) = 1 | − 2·2-s − 3-s + 2·4-s − 5-s + 2·6-s − 2·7-s + 9-s + 2·10-s + 11-s − 2·12-s + 2·13-s + 4·14-s + 15-s − 4·16-s + 2·17-s − 2·18-s − 2·20-s + 2·21-s − 2·22-s − 4·23-s + 25-s − 4·26-s − 27-s − 4·28-s − 5·29-s − 2·30-s + 9·31-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 4-s − 0.447·5-s + 0.816·6-s − 0.755·7-s + 1/3·9-s + 0.632·10-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s − 16-s + 0.485·17-s − 0.471·18-s − 0.447·20-s + 0.436·21-s − 0.426·22-s − 0.834·23-s + 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.755·28-s − 0.928·29-s − 0.365·30-s + 1.61·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−7.992131401491151341853022691834, −7.19943312357885437312744870075, −6.57076939524989312227924697486, −5.97393760565884410772576328994, −4.92867898754526378757486525883, −4.03015791710918251989494267582, −3.21996156514345266704871794211, −1.95238718439483675366934989313, −0.963933067556842158966374204033, 0,
0.963933067556842158966374204033, 1.95238718439483675366934989313, 3.21996156514345266704871794211, 4.03015791710918251989494267582, 4.92867898754526378757486525883, 5.97393760565884410772576328994, 6.57076939524989312227924697486, 7.19943312357885437312744870075, 7.992131401491151341853022691834