| L(s) = 1 | + 2.61·2-s + 4.85·4-s + 1.38·5-s − 4.23·7-s + 7.47·8-s + 3.61·10-s + 1.76·11-s + 2.85·13-s − 11.0·14-s + 9.85·16-s + 4.23·17-s − 6.23·19-s + 6.70·20-s + 4.61·22-s + 4.85·23-s − 3.09·25-s + 7.47·26-s − 20.5·28-s − 9.32·29-s − 0.472·31-s + 10.8·32-s + 11.0·34-s − 5.85·35-s − 6.09·37-s − 16.3·38-s + 10.3·40-s + 7.38·41-s + ⋯ |
| L(s) = 1 | + 1.85·2-s + 2.42·4-s + 0.618·5-s − 1.60·7-s + 2.64·8-s + 1.14·10-s + 0.531·11-s + 0.791·13-s − 2.96·14-s + 2.46·16-s + 1.02·17-s − 1.43·19-s + 1.49·20-s + 0.984·22-s + 1.01·23-s − 0.618·25-s + 1.46·26-s − 3.88·28-s − 1.73·29-s − 0.0847·31-s + 1.91·32-s + 1.90·34-s − 0.989·35-s − 1.00·37-s − 2.64·38-s + 1.63·40-s + 1.15·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.333928097\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.333928097\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + T \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 5 | \( 1 - 1.38T + 5T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 1.76T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 4.23T + 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 - 4.85T + 23T^{2} \) |
| 29 | \( 1 + 9.32T + 29T^{2} \) |
| 31 | \( 1 + 0.472T + 31T^{2} \) |
| 37 | \( 1 + 6.09T + 37T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 4.14T + 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 + 13.1T + 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 + 1.38T + 79T^{2} \) |
| 83 | \( 1 - 8.23T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−10.81288353350062675684502972297, −9.935505011719344266310064308623, −8.991985429707588963801107464720, −7.42875850799259660731294530678, −6.34851534825656300412733144074, −6.17929430481680381305525766339, −5.12552713637516624106259395067, −3.76495652252405244172700511352, −3.30296231593470578555090815355, −1.94130245439865858405256863114,
1.94130245439865858405256863114, 3.30296231593470578555090815355, 3.76495652252405244172700511352, 5.12552713637516624106259395067, 6.17929430481680381305525766339, 6.34851534825656300412733144074, 7.42875850799259660731294530678, 8.991985429707588963801107464720, 9.935505011719344266310064308623, 10.81288353350062675684502972297
