L(s) = 1 | + 2-s − 2.77·3-s + 4-s − 5-s − 2.77·6-s + 7-s + 8-s + 4.71·9-s − 10-s − 2.77·12-s − 2.77·13-s + 14-s + 2.77·15-s + 16-s + 6.71·17-s + 4.71·18-s + 7.49·19-s − 20-s − 2.77·21-s − 3.83·23-s − 2.77·24-s + 25-s − 2.77·26-s − 4.77·27-s + 28-s − 5.43·29-s + 2.77·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.60·3-s + 0.5·4-s − 0.447·5-s − 1.13·6-s + 0.377·7-s + 0.353·8-s + 1.57·9-s − 0.316·10-s − 0.802·12-s − 0.770·13-s + 0.267·14-s + 0.717·15-s + 0.250·16-s + 1.62·17-s + 1.11·18-s + 1.72·19-s − 0.223·20-s − 0.606·21-s − 0.800·23-s − 0.567·24-s + 0.200·25-s − 0.544·26-s − 0.919·27-s + 0.188·28-s − 1.01·29-s + 0.507·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.760175120\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.760175120\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 + 2.77T + 3T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 - 6.71T + 17T^{2} \) |
| 19 | \( 1 - 7.49T + 19T^{2} \) |
| 23 | \( 1 + 3.83T + 23T^{2} \) |
| 29 | \( 1 + 5.43T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 0.117T + 37T^{2} \) |
| 41 | \( 1 - 5.55T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 2.11T + 47T^{2} \) |
| 53 | \( 1 + 2.71T + 53T^{2} \) |
| 59 | \( 1 + 3.49T + 59T^{2} \) |
| 61 | \( 1 + 4.27T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 3.16T + 71T^{2} \) |
| 73 | \( 1 - 8.83T + 73T^{2} \) |
| 79 | \( 1 + 8.55T + 79T^{2} \) |
| 83 | \( 1 + 0.778T + 83T^{2} \) |
| 89 | \( 1 - 9.55T + 89T^{2} \) |
| 97 | \( 1 + 4.83T + 97T^{2} \) |
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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
−7.41796634582373507851988256063, −7.21663927365504825662474436251, −5.99793704410532733246358230779, −5.71592395767365072962627520909, −5.11911930246173555734499484434, −4.48236543912557344994949318765, −3.70893360839727664214844433337, −2.82744891943637132716997706387, −1.53550203288305370453418088176, −0.67335676781879113890718087737,
0.67335676781879113890718087737, 1.53550203288305370453418088176, 2.82744891943637132716997706387, 3.70893360839727664214844433337, 4.48236543912557344994949318765, 5.11911930246173555734499484434, 5.71592395767365072962627520909, 5.99793704410532733246358230779, 7.21663927365504825662474436251, 7.41796634582373507851988256063