| L(s) = 1 | + 0.523·2-s − 1.72·4-s + 5-s − 3.20·7-s − 1.95·8-s + 0.523·10-s + 5.20·11-s + 2·13-s − 1.67·14-s + 2.42·16-s − 17-s + 7.20·19-s − 1.72·20-s + 2.72·22-s + 25-s + 1.04·26-s + 5.52·28-s − 0.249·29-s + 6.40·31-s + 5.17·32-s − 0.523·34-s − 3.20·35-s + 2.24·37-s + 3.77·38-s − 1.95·40-s + 10.6·41-s + 4.09·43-s + ⋯ |
| L(s) = 1 | + 0.370·2-s − 0.862·4-s + 0.447·5-s − 1.21·7-s − 0.690·8-s + 0.165·10-s + 1.56·11-s + 0.554·13-s − 0.448·14-s + 0.607·16-s − 0.242·17-s + 1.65·19-s − 0.385·20-s + 0.581·22-s + 0.200·25-s + 0.205·26-s + 1.04·28-s − 0.0463·29-s + 1.15·31-s + 0.915·32-s − 0.0898·34-s − 0.541·35-s + 0.369·37-s + 0.612·38-s − 0.308·40-s + 1.66·41-s + 0.624·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 765 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.553034565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.553034565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 2 | \( 1 - 0.523T + 2T^{2} \) |
| 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 - 5.20T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 7.20T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 0.249T + 29T^{2} \) |
| 31 | \( 1 - 6.40T + 31T^{2} \) |
| 37 | \( 1 - 2.24T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 - 4.09T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 1.84T + 53T^{2} \) |
| 59 | \( 1 + 10.4T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 6.49T + 71T^{2} \) |
| 73 | \( 1 - 3.84T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 8.80T + 83T^{2} \) |
| 89 | \( 1 - 8.30T + 89T^{2} \) |
| 97 | \( 1 + 4.49T + 97T^{2} \) |
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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
−9.996507636681585870196186158671, −9.380629891877635518562152066488, −9.033196747275986115057555718453, −7.75542974267860849815779665646, −6.42448569754507790529843127470, −6.08923928799591751993798628327, −4.84908123140523207157713403509, −3.80592726158348554010999093982, −3.05027514386693319965995463758, −1.04776292674763783584521470963,
1.04776292674763783584521470963, 3.05027514386693319965995463758, 3.80592726158348554010999093982, 4.84908123140523207157713403509, 6.08923928799591751993798628327, 6.42448569754507790529843127470, 7.75542974267860849815779665646, 9.033196747275986115057555718453, 9.380629891877635518562152066488, 9.996507636681585870196186158671
