L(s) = 1 | + 1.61·2-s + 0.618·4-s − 3.23·5-s + 1.23·7-s − 2.23·8-s − 5.23·10-s − 0.763·11-s + 3·13-s + 2.00·14-s − 4.85·16-s + 5.23·17-s + 2·19-s − 2.00·20-s − 1.23·22-s + 5.47·25-s + 4.85·26-s + 0.763·28-s + 3·29-s − 6.70·31-s − 3.38·32-s + 8.47·34-s − 4.00·35-s − 3.23·37-s + 3.23·38-s + 7.23·40-s − 5.47·41-s − 0.472·44-s + ⋯ |
L(s) = 1 | + 1.14·2-s + 0.309·4-s − 1.44·5-s + 0.467·7-s − 0.790·8-s − 1.65·10-s − 0.230·11-s + 0.832·13-s + 0.534·14-s − 1.21·16-s + 1.26·17-s + 0.458·19-s − 0.447·20-s − 0.263·22-s + 1.09·25-s + 0.951·26-s + 0.144·28-s + 0.557·29-s − 1.20·31-s − 0.597·32-s + 1.45·34-s − 0.676·35-s − 0.532·37-s + 0.524·38-s + 1.14·40-s − 0.854·41-s − 0.0711·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4761 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 - 5.23T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + 6.70T + 31T^{2} \) |
| 37 | \( 1 + 3.23T + 37T^{2} \) |
| 41 | \( 1 + 5.47T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 2.23T + 47T^{2} \) |
| 53 | \( 1 + 8.47T + 53T^{2} \) |
| 59 | \( 1 - 2.47T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 + 7.76T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 6.94T + 79T^{2} \) |
| 83 | \( 1 + 13.2T + 83T^{2} \) |
| 89 | \( 1 + 1.52T + 89T^{2} \) |
| 97 | \( 1 + 4.29T + 97T^{2} \) |
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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
−7.949023216455438283082351834839, −7.18779811222303166428278141409, −6.34991964220925961774885939788, −5.43895753897615430157894028882, −4.94577275874817118244622519038, −4.08232764053907370732480318652, −3.52574331967385892167828632452, −2.96970095968514349897921819551, −1.40696769912048394010479426712, 0,
1.40696769912048394010479426712, 2.96970095968514349897921819551, 3.52574331967385892167828632452, 4.08232764053907370732480318652, 4.94577275874817118244622519038, 5.43895753897615430157894028882, 6.34991964220925961774885939788, 7.18779811222303166428278141409, 7.949023216455438283082351834839