| L(s) = 1 | + 1.14·3-s − 5-s + 3.63·7-s − 1.69·9-s + 1.42·11-s − 5.50·13-s − 1.14·15-s − 3.01·17-s − 2.20·19-s + 4.14·21-s − 6.84·23-s + 25-s − 5.36·27-s + 6.35·29-s − 7.54·31-s + 1.62·33-s − 3.63·35-s − 37-s − 6.28·39-s − 8.54·41-s − 4.05·43-s + 1.69·45-s + 10.4·47-s + 6.18·49-s − 3.44·51-s − 8.13·53-s − 1.42·55-s + ⋯ |
| L(s) = 1 | + 0.659·3-s − 0.447·5-s + 1.37·7-s − 0.564·9-s + 0.428·11-s − 1.52·13-s − 0.295·15-s − 0.730·17-s − 0.506·19-s + 0.905·21-s − 1.42·23-s + 0.200·25-s − 1.03·27-s + 1.18·29-s − 1.35·31-s + 0.282·33-s − 0.613·35-s − 0.164·37-s − 1.00·39-s − 1.33·41-s − 0.618·43-s + 0.252·45-s + 1.51·47-s + 0.883·49-s − 0.482·51-s − 1.11·53-s − 0.191·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 3 | \( 1 - 1.14T + 3T^{2} \) |
| 7 | \( 1 - 3.63T + 7T^{2} \) |
| 11 | \( 1 - 1.42T + 11T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 17 | \( 1 + 3.01T + 17T^{2} \) |
| 19 | \( 1 + 2.20T + 19T^{2} \) |
| 23 | \( 1 + 6.84T + 23T^{2} \) |
| 29 | \( 1 - 6.35T + 29T^{2} \) |
| 31 | \( 1 + 7.54T + 31T^{2} \) |
| 41 | \( 1 + 8.54T + 41T^{2} \) |
| 43 | \( 1 + 4.05T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 + 8.13T + 53T^{2} \) |
| 59 | \( 1 + 1.23T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 + 7.81T + 67T^{2} \) |
| 71 | \( 1 - 0.806T + 71T^{2} \) |
| 73 | \( 1 - 2.80T + 73T^{2} \) |
| 79 | \( 1 + 0.754T + 79T^{2} \) |
| 83 | \( 1 + 7.41T + 83T^{2} \) |
| 89 | \( 1 - 14.8T + 89T^{2} \) |
| 97 | \( 1 + 5.94T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.406907025667607557332877667435, −7.73695631471981098094609372167, −7.12750633048717467773082147242, −6.08099790361919706753028675630, −5.04946693042311981467179011070, −4.48952614849450982971403825915, −3.58755565249821715744173356446, −2.44198459317863686744156946864, −1.80838948743550493535863936147, 0,
1.80838948743550493535863936147, 2.44198459317863686744156946864, 3.58755565249821715744173356446, 4.48952614849450982971403825915, 5.04946693042311981467179011070, 6.08099790361919706753028675630, 7.12750633048717467773082147242, 7.73695631471981098094609372167, 8.406907025667607557332877667435
