L(s) = 1 | + 2-s − 1.30·3-s + 4-s − 5-s − 1.30·6-s + 0.511·7-s + 8-s − 1.30·9-s − 10-s − 2.76·11-s − 1.30·12-s + 1.86·13-s + 0.511·14-s + 1.30·15-s + 16-s + 2.86·17-s − 1.30·18-s − 5.33·19-s − 20-s − 0.666·21-s − 2.76·22-s + 6.87·23-s − 1.30·24-s + 25-s + 1.86·26-s + 5.60·27-s + 0.511·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.751·3-s + 0.5·4-s − 0.447·5-s − 0.531·6-s + 0.193·7-s + 0.353·8-s − 0.435·9-s − 0.316·10-s − 0.834·11-s − 0.375·12-s + 0.517·13-s + 0.136·14-s + 0.336·15-s + 0.250·16-s + 0.693·17-s − 0.307·18-s − 1.22·19-s − 0.223·20-s − 0.145·21-s − 0.589·22-s + 1.43·23-s − 0.265·24-s + 0.200·25-s + 0.366·26-s + 1.07·27-s + 0.0966·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 7 | \( 1 - 0.511T + 7T^{2} \) |
| 11 | \( 1 + 2.76T + 11T^{2} \) |
| 13 | \( 1 - 1.86T + 13T^{2} \) |
| 17 | \( 1 - 2.86T + 17T^{2} \) |
| 19 | \( 1 + 5.33T + 19T^{2} \) |
| 23 | \( 1 - 6.87T + 23T^{2} \) |
| 29 | \( 1 + 0.778T + 29T^{2} \) |
| 31 | \( 1 + 4.45T + 31T^{2} \) |
| 37 | \( 1 - 9.31T + 37T^{2} \) |
| 41 | \( 1 + 6.78T + 41T^{2} \) |
| 43 | \( 1 - 7.76T + 43T^{2} \) |
| 47 | \( 1 + 3.86T + 47T^{2} \) |
| 53 | \( 1 + 13.5T + 53T^{2} \) |
| 59 | \( 1 - 5.31T + 59T^{2} \) |
| 61 | \( 1 - 9.06T + 61T^{2} \) |
| 67 | \( 1 - 0.425T + 67T^{2} \) |
| 71 | \( 1 + 5.77T + 71T^{2} \) |
| 73 | \( 1 + 5.03T + 73T^{2} \) |
| 79 | \( 1 + 8.28T + 79T^{2} \) |
| 83 | \( 1 + 1.66T + 83T^{2} \) |
| 89 | \( 1 + 3.59T + 89T^{2} \) |
| 97 | \( 1 + 4.07T + 97T^{2} \) |
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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
−7.67669127159722721741002775931, −6.81377026348240129122101055808, −6.18974347270850589527258711782, −5.47805962976538595314983689344, −4.95133993096299270300119338903, −4.19710196752723285863733485117, −3.25574372672874655626449706115, −2.57063785265181275617399602819, −1.28024876948149895418104140227, 0,
1.28024876948149895418104140227, 2.57063785265181275617399602819, 3.25574372672874655626449706115, 4.19710196752723285863733485117, 4.95133993096299270300119338903, 5.47805962976538595314983689344, 6.18974347270850589527258711782, 6.81377026348240129122101055808, 7.67669127159722721741002775931