| L(s) = 1 | + 2-s + 0.289·3-s + 4-s − 5-s + 0.289·6-s + 7-s + 8-s − 2.91·9-s − 10-s + 0.289·12-s + 0.289·13-s + 14-s − 0.289·15-s + 16-s − 0.916·17-s − 2.91·18-s − 3.20·19-s − 20-s + 0.289·21-s − 5.33·23-s + 0.289·24-s + 25-s + 0.289·26-s − 1.71·27-s + 28-s + 9.83·29-s − 0.289·30-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.166·3-s + 0.5·4-s − 0.447·5-s + 0.118·6-s + 0.377·7-s + 0.353·8-s − 0.972·9-s − 0.316·10-s + 0.0834·12-s + 0.0802·13-s + 0.267·14-s − 0.0746·15-s + 0.250·16-s − 0.222·17-s − 0.687·18-s − 0.735·19-s − 0.223·20-s + 0.0631·21-s − 1.11·23-s + 0.0590·24-s + 0.200·25-s + 0.0567·26-s − 0.329·27-s + 0.188·28-s + 1.82·29-s − 0.0527·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.799757082\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.799757082\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - 0.289T + 3T^{2} \) |
| 13 | \( 1 - 0.289T + 13T^{2} \) |
| 17 | \( 1 + 0.916T + 17T^{2} \) |
| 19 | \( 1 + 3.20T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 - 9.83T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 9.25T + 37T^{2} \) |
| 41 | \( 1 + 0.578T + 41T^{2} \) |
| 43 | \( 1 - 5.75T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 59 | \( 1 - 7.20T + 59T^{2} \) |
| 61 | \( 1 - 9.49T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 - 10.3T + 73T^{2} \) |
| 79 | \( 1 + 2.42T + 79T^{2} \) |
| 83 | \( 1 - 2.28T + 83T^{2} \) |
| 89 | \( 1 - 3.42T + 89T^{2} \) |
| 97 | \( 1 + 6.33T + 97T^{2} \) |
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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
−7.82317464091137713829717402126, −6.97527323139160506707512202520, −6.38067569494574578714195185231, −5.61545770335549493674546235921, −5.01914665972444400290957349314, −4.14598666212608247994587603083, −3.66333959878010799718416476833, −2.64440171493433481428624307734, −2.10459948107619394671808123086, −0.70622398268489978735753856046,
0.70622398268489978735753856046, 2.10459948107619394671808123086, 2.64440171493433481428624307734, 3.66333959878010799718416476833, 4.14598666212608247994587603083, 5.01914665972444400290957349314, 5.61545770335549493674546235921, 6.38067569494574578714195185231, 6.97527323139160506707512202520, 7.82317464091137713829717402126
