| L(s) = 1 | − 1.90·2-s − 3-s + 1.61·4-s − 5-s + 1.90·6-s − 0.941·7-s + 0.732·8-s + 9-s + 1.90·10-s − 0.344·11-s − 1.61·12-s − 3.84·13-s + 1.79·14-s + 15-s − 4.62·16-s − 0.110·17-s − 1.90·18-s − 6.17·19-s − 1.61·20-s + 0.941·21-s + 0.655·22-s − 0.732·24-s + 25-s + 7.31·26-s − 27-s − 1.52·28-s − 9.60·29-s + ⋯ |
| L(s) = 1 | − 1.34·2-s − 0.577·3-s + 0.807·4-s − 0.447·5-s + 0.776·6-s − 0.355·7-s + 0.258·8-s + 0.333·9-s + 0.601·10-s − 0.103·11-s − 0.466·12-s − 1.06·13-s + 0.478·14-s + 0.258·15-s − 1.15·16-s − 0.0269·17-s − 0.448·18-s − 1.41·19-s − 0.361·20-s + 0.205·21-s + 0.139·22-s − 0.149·24-s + 0.200·25-s + 1.43·26-s − 0.192·27-s − 0.287·28-s − 1.78·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.02980394498\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02980394498\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + 1.90T + 2T^{2} \) |
| 7 | \( 1 + 0.941T + 7T^{2} \) |
| 11 | \( 1 + 0.344T + 11T^{2} \) |
| 13 | \( 1 + 3.84T + 13T^{2} \) |
| 17 | \( 1 + 0.110T + 17T^{2} \) |
| 19 | \( 1 + 6.17T + 19T^{2} \) |
| 29 | \( 1 + 9.60T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 + 5.89T + 37T^{2} \) |
| 41 | \( 1 - 7.16T + 41T^{2} \) |
| 43 | \( 1 + 0.975T + 43T^{2} \) |
| 47 | \( 1 + 2.28T + 47T^{2} \) |
| 53 | \( 1 + 13.4T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 2.16T + 61T^{2} \) |
| 67 | \( 1 - 2.89T + 67T^{2} \) |
| 71 | \( 1 - 0.0848T + 71T^{2} \) |
| 73 | \( 1 + 2.32T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 + 4.58T + 83T^{2} \) |
| 89 | \( 1 + 11.8T + 89T^{2} \) |
| 97 | \( 1 - 13.6T + 97T^{2} \) |
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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
−7.78127702559875551776569290280, −7.37878932834256599693916344339, −6.68840169452376878878008815788, −5.99045953004011019259136717156, −4.98399187780640015630067958104, −4.41230486232608838690437065362, −3.47896322115129953108364752767, −2.30488987047572401561563117158, −1.54333251055780298569733819625, −0.10825684376617065299394319851,
0.10825684376617065299394319851, 1.54333251055780298569733819625, 2.30488987047572401561563117158, 3.47896322115129953108364752767, 4.41230486232608838690437065362, 4.98399187780640015630067958104, 5.99045953004011019259136717156, 6.68840169452376878878008815788, 7.37878932834256599693916344339, 7.78127702559875551776569290280
