| L(s) = 1 | + 0.618·2-s − 1.61·4-s + 0.381·5-s + 7-s − 2.23·8-s + 0.236·10-s − 13-s + 0.618·14-s + 1.85·16-s + 4.23·17-s − 0.618·20-s + 3.23·23-s − 4.85·25-s − 0.618·26-s − 1.61·28-s − 6.70·29-s − 10.2·31-s + 5.61·32-s + 2.61·34-s + 0.381·35-s + 6.94·37-s − 0.854·40-s − 5.09·41-s − 43-s + 2.00·46-s + 7.32·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s + 0.170·5-s + 0.377·7-s − 0.790·8-s + 0.0746·10-s − 0.277·13-s + 0.165·14-s + 0.463·16-s + 1.02·17-s − 0.138·20-s + 0.674·23-s − 0.970·25-s − 0.121·26-s − 0.305·28-s − 1.24·29-s − 1.83·31-s + 0.993·32-s + 0.448·34-s + 0.0645·35-s + 1.14·37-s − 0.135·40-s − 0.794·41-s − 0.152·43-s + 0.294·46-s + 1.06·47-s + 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 - 0.381T + 5T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + 10.2T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 + 5.09T + 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 - 7.32T + 47T^{2} \) |
| 53 | \( 1 + 7.61T + 53T^{2} \) |
| 59 | \( 1 - 4.14T + 59T^{2} \) |
| 61 | \( 1 + 5.76T + 61T^{2} \) |
| 67 | \( 1 + 9.23T + 67T^{2} \) |
| 71 | \( 1 - 7.47T + 71T^{2} \) |
| 73 | \( 1 - 11.5T + 73T^{2} \) |
| 79 | \( 1 - 10.8T + 79T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 - 6.38T + 89T^{2} \) |
| 97 | \( 1 + 17T + 97T^{2} \) |
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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
−7.74607574496326732163732655815, −6.80293717043010550500974724616, −5.79973212324884226975447910529, −5.44421402194162496616611601776, −4.77250187209593489514959980178, −3.87219579206636883680054317876, −3.40356513863813462369537628487, −2.30442816032702091542250214980, −1.26301442173657144658398263786, 0,
1.26301442173657144658398263786, 2.30442816032702091542250214980, 3.40356513863813462369537628487, 3.87219579206636883680054317876, 4.77250187209593489514959980178, 5.44421402194162496616611601776, 5.79973212324884226975447910529, 6.80293717043010550500974724616, 7.74607574496326732163732655815
