L(s) = 1 | + 1.89·2-s − 1.86·3-s + 1.59·4-s + 5-s − 3.53·6-s + 0.586·7-s − 0.760·8-s + 0.471·9-s + 1.89·10-s + 11-s − 2.97·12-s + 1.52·13-s + 1.11·14-s − 1.86·15-s − 4.64·16-s + 0.519·17-s + 0.893·18-s − 5.41·19-s + 1.59·20-s − 1.09·21-s + 1.89·22-s − 1.24·23-s + 1.41·24-s + 25-s + 2.88·26-s + 4.71·27-s + 0.938·28-s + ⋯ |
L(s) = 1 | + 1.34·2-s − 1.07·3-s + 0.799·4-s + 0.447·5-s − 1.44·6-s + 0.221·7-s − 0.269·8-s + 0.157·9-s + 0.599·10-s + 0.301·11-s − 0.859·12-s + 0.422·13-s + 0.297·14-s − 0.481·15-s − 1.16·16-s + 0.125·17-s + 0.210·18-s − 1.24·19-s + 0.357·20-s − 0.238·21-s + 0.404·22-s − 0.259·23-s + 0.289·24-s + 0.200·25-s + 0.566·26-s + 0.906·27-s + 0.177·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 73 | \( 1 + T \) |
good | 2 | \( 1 - 1.89T + 2T^{2} \) |
| 3 | \( 1 + 1.86T + 3T^{2} \) |
| 7 | \( 1 - 0.586T + 7T^{2} \) |
| 13 | \( 1 - 1.52T + 13T^{2} \) |
| 17 | \( 1 - 0.519T + 17T^{2} \) |
| 19 | \( 1 + 5.41T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 + 0.277T + 29T^{2} \) |
| 31 | \( 1 - 3.17T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 - 9.94T + 41T^{2} \) |
| 43 | \( 1 - 4.36T + 43T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + 2.87T + 53T^{2} \) |
| 59 | \( 1 + 15.0T + 59T^{2} \) |
| 61 | \( 1 + 6.51T + 61T^{2} \) |
| 67 | \( 1 - 0.812T + 67T^{2} \) |
| 71 | \( 1 + 4.33T + 71T^{2} \) |
| 79 | \( 1 - 10.5T + 79T^{2} \) |
| 83 | \( 1 - 16.0T + 83T^{2} \) |
| 89 | \( 1 + 11.6T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−7.998010003409708953451943889521, −6.78529357283957837803062558534, −6.27084195718936524874714274031, −5.86450931303877244041227049945, −5.01493543250252870106132345303, −4.54730525480421397585651632528, −3.64383302488682712279955728816, −2.69944579307661320918234369613, −1.56574367832079550116499603808, 0,
1.56574367832079550116499603808, 2.69944579307661320918234369613, 3.64383302488682712279955728816, 4.54730525480421397585651632528, 5.01493543250252870106132345303, 5.86450931303877244041227049945, 6.27084195718936524874714274031, 6.78529357283957837803062558534, 7.998010003409708953451943889521