L(s) = 1 | + 2-s + 1.58·3-s + 4-s − 5-s + 1.58·6-s + 0.389·7-s + 8-s − 0.500·9-s − 10-s − 4.95·11-s + 1.58·12-s − 1.44·13-s + 0.389·14-s − 1.58·15-s + 16-s + 2.82·17-s − 0.500·18-s − 1.72·19-s − 20-s + 0.615·21-s − 4.95·22-s − 4.93·23-s + 1.58·24-s + 25-s − 1.44·26-s − 5.53·27-s + 0.389·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.912·3-s + 0.5·4-s − 0.447·5-s + 0.645·6-s + 0.147·7-s + 0.353·8-s − 0.166·9-s − 0.316·10-s − 1.49·11-s + 0.456·12-s − 0.400·13-s + 0.104·14-s − 0.408·15-s + 0.250·16-s + 0.686·17-s − 0.117·18-s − 0.396·19-s − 0.223·20-s + 0.134·21-s − 1.05·22-s − 1.02·23-s + 0.322·24-s + 0.200·25-s − 0.283·26-s − 1.06·27-s + 0.0735·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 - 1.58T + 3T^{2} \) |
| 7 | \( 1 - 0.389T + 7T^{2} \) |
| 11 | \( 1 + 4.95T + 11T^{2} \) |
| 13 | \( 1 + 1.44T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 + 4.93T + 23T^{2} \) |
| 29 | \( 1 + 2.52T + 29T^{2} \) |
| 31 | \( 1 + 7.36T + 31T^{2} \) |
| 37 | \( 1 + 7.33T + 37T^{2} \) |
| 41 | \( 1 - 7.73T + 41T^{2} \) |
| 43 | \( 1 - 1.27T + 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 + 3.41T + 53T^{2} \) |
| 59 | \( 1 - 8.75T + 59T^{2} \) |
| 61 | \( 1 - 1.75T + 61T^{2} \) |
| 67 | \( 1 - 12.8T + 67T^{2} \) |
| 71 | \( 1 - 3.84T + 71T^{2} \) |
| 73 | \( 1 + 3.04T + 73T^{2} \) |
| 79 | \( 1 - 2.18T + 79T^{2} \) |
| 83 | \( 1 - 12.5T + 83T^{2} \) |
| 89 | \( 1 + 17.2T + 89T^{2} \) |
| 97 | \( 1 - 3.18T + 97T^{2} \) |
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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
−7.86882321526380463131208654011, −7.66130032040334515934979349256, −6.65832490841830516217165838233, −5.54493085430131566329008090758, −5.19511205991552847930382245966, −4.06369333889019111539470157438, −3.42709244404427435325919330532, −2.61679890559348993466804564039, −1.91738838178545038911104912942, 0,
1.91738838178545038911104912942, 2.61679890559348993466804564039, 3.42709244404427435325919330532, 4.06369333889019111539470157438, 5.19511205991552847930382245966, 5.54493085430131566329008090758, 6.65832490841830516217165838233, 7.66130032040334515934979349256, 7.86882321526380463131208654011