L(s) = 1 | − 2-s + 2.63·3-s + 4-s − 1.57·5-s − 2.63·6-s − 2.35·7-s − 8-s + 3.93·9-s + 1.57·10-s + 2.96·11-s + 2.63·12-s + 2.20·13-s + 2.35·14-s − 4.14·15-s + 16-s − 0.149·17-s − 3.93·18-s − 4.54·19-s − 1.57·20-s − 6.19·21-s − 2.96·22-s + 23-s − 2.63·24-s − 2.52·25-s − 2.20·26-s + 2.47·27-s − 2.35·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1.52·3-s + 0.5·4-s − 0.703·5-s − 1.07·6-s − 0.888·7-s − 0.353·8-s + 1.31·9-s + 0.497·10-s + 0.894·11-s + 0.760·12-s + 0.611·13-s + 0.628·14-s − 1.06·15-s + 0.250·16-s − 0.0362·17-s − 0.928·18-s − 1.04·19-s − 0.351·20-s − 1.35·21-s − 0.632·22-s + 0.208·23-s − 0.537·24-s − 0.505·25-s − 0.432·26-s + 0.475·27-s − 0.444·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 2.63T + 3T^{2} \) |
| 5 | \( 1 + 1.57T + 5T^{2} \) |
| 7 | \( 1 + 2.35T + 7T^{2} \) |
| 11 | \( 1 - 2.96T + 11T^{2} \) |
| 13 | \( 1 - 2.20T + 13T^{2} \) |
| 17 | \( 1 + 0.149T + 17T^{2} \) |
| 19 | \( 1 + 4.54T + 19T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 - 7.19T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 4.11T + 53T^{2} \) |
| 59 | \( 1 + 1.51T + 59T^{2} \) |
| 61 | \( 1 - 8.49T + 61T^{2} \) |
| 67 | \( 1 + 3.65T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 9.06T + 73T^{2} \) |
| 79 | \( 1 - 3.04T + 79T^{2} \) |
| 83 | \( 1 + 4.50T + 83T^{2} \) |
| 89 | \( 1 + 7.58T + 89T^{2} \) |
| 97 | \( 1 + 6.63T + 97T^{2} \) |
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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
−8.067153392157615364658398565741, −7.04206066536716140039549484345, −6.75148455594140969576808769217, −5.83778893391511023968610941470, −4.45888202643597090457311292478, −3.59760632577411631138183631181, −3.37428724303693409102492053975, −2.30645477896050177829763866439, −1.46274412810410308467676941216, 0,
1.46274412810410308467676941216, 2.30645477896050177829763866439, 3.37428724303693409102492053975, 3.59760632577411631138183631181, 4.45888202643597090457311292478, 5.83778893391511023968610941470, 6.75148455594140969576808769217, 7.04206066536716140039549484345, 8.067153392157615364658398565741