L(s) = 1 | − 2.70·3-s + 2.99·5-s − 2.81·7-s + 4.34·9-s − 3.12·11-s + 1.74·13-s − 8.11·15-s + 0.354·17-s − 0.768·19-s + 7.61·21-s + 2.62·23-s + 3.97·25-s − 3.63·27-s − 2.66·29-s + 7.34·31-s + 8.46·33-s − 8.41·35-s − 5.78·37-s − 4.71·39-s − 6.53·41-s + 11.2·43-s + 13.0·45-s − 11.5·47-s + 0.897·49-s − 0.960·51-s + 5.59·53-s − 9.35·55-s + ⋯ |
L(s) = 1 | − 1.56·3-s + 1.33·5-s − 1.06·7-s + 1.44·9-s − 0.941·11-s + 0.482·13-s − 2.09·15-s + 0.0859·17-s − 0.176·19-s + 1.66·21-s + 0.547·23-s + 0.794·25-s − 0.700·27-s − 0.494·29-s + 1.31·31-s + 1.47·33-s − 1.42·35-s − 0.950·37-s − 0.755·39-s − 1.02·41-s + 1.70·43-s + 1.93·45-s − 1.68·47-s + 0.128·49-s − 0.134·51-s + 0.768·53-s − 1.26·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
good | 3 | \( 1 + 2.70T + 3T^{2} \) |
| 5 | \( 1 - 2.99T + 5T^{2} \) |
| 7 | \( 1 + 2.81T + 7T^{2} \) |
| 11 | \( 1 + 3.12T + 11T^{2} \) |
| 13 | \( 1 - 1.74T + 13T^{2} \) |
| 17 | \( 1 - 0.354T + 17T^{2} \) |
| 19 | \( 1 + 0.768T + 19T^{2} \) |
| 23 | \( 1 - 2.62T + 23T^{2} \) |
| 29 | \( 1 + 2.66T + 29T^{2} \) |
| 31 | \( 1 - 7.34T + 31T^{2} \) |
| 37 | \( 1 + 5.78T + 37T^{2} \) |
| 41 | \( 1 + 6.53T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 11.5T + 47T^{2} \) |
| 53 | \( 1 - 5.59T + 53T^{2} \) |
| 59 | \( 1 + 5.94T + 59T^{2} \) |
| 61 | \( 1 - 13.2T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 + 1.16T + 71T^{2} \) |
| 73 | \( 1 - 7.40T + 73T^{2} \) |
| 79 | \( 1 - 1.11T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 89 | \( 1 - 8.47T + 89T^{2} \) |
| 97 | \( 1 + 11.3T + 97T^{2} \) |
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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
−8.016601672123811884528156228971, −6.85717812922946653219504769817, −6.51291702059053784375637274628, −5.79532319812295683534518234413, −5.38217680407997825376140465478, −4.61538561202636721773726728679, −3.36152086137341631286116464933, −2.37693239783064799291715364838, −1.20236544172396326431578701004, 0,
1.20236544172396326431578701004, 2.37693239783064799291715364838, 3.36152086137341631286116464933, 4.61538561202636721773726728679, 5.38217680407997825376140465478, 5.79532319812295683534518234413, 6.51291702059053784375637274628, 6.85717812922946653219504769817, 8.016601672123811884528156228971