| L(s) = 1 | + 0.264·2-s + 3-s − 1.92·4-s + 5-s + 0.264·6-s + 0.526·7-s − 1.03·8-s + 9-s + 0.264·10-s + 0.0609·11-s − 1.92·12-s − 1.26·13-s + 0.139·14-s + 15-s + 3.58·16-s + 1.96·17-s + 0.264·18-s − 6.64·19-s − 1.92·20-s + 0.526·21-s + 0.0161·22-s − 1.03·24-s + 25-s − 0.333·26-s + 27-s − 1.01·28-s + 3.75·29-s + ⋯ |
| L(s) = 1 | + 0.187·2-s + 0.577·3-s − 0.964·4-s + 0.447·5-s + 0.108·6-s + 0.198·7-s − 0.367·8-s + 0.333·9-s + 0.0836·10-s + 0.0183·11-s − 0.557·12-s − 0.349·13-s + 0.0372·14-s + 0.258·15-s + 0.896·16-s + 0.477·17-s + 0.0623·18-s − 1.52·19-s − 0.431·20-s + 0.114·21-s + 0.00343·22-s − 0.212·24-s + 0.200·25-s − 0.0654·26-s + 0.192·27-s − 0.191·28-s + 0.697·29-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 - 0.264T + 2T^{2} \) |
| 7 | \( 1 - 0.526T + 7T^{2} \) |
| 11 | \( 1 - 0.0609T + 11T^{2} \) |
| 13 | \( 1 + 1.26T + 13T^{2} \) |
| 17 | \( 1 - 1.96T + 17T^{2} \) |
| 19 | \( 1 + 6.64T + 19T^{2} \) |
| 29 | \( 1 - 3.75T + 29T^{2} \) |
| 31 | \( 1 + 8.07T + 31T^{2} \) |
| 37 | \( 1 - 0.263T + 37T^{2} \) |
| 41 | \( 1 - 2.87T + 41T^{2} \) |
| 43 | \( 1 - 4.13T + 43T^{2} \) |
| 47 | \( 1 + 9.43T + 47T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 59 | \( 1 + 4.16T + 59T^{2} \) |
| 61 | \( 1 + 2.68T + 61T^{2} \) |
| 67 | \( 1 + 10.5T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 + 1.37T + 73T^{2} \) |
| 79 | \( 1 - 2.33T + 79T^{2} \) |
| 83 | \( 1 - 0.155T + 83T^{2} \) |
| 89 | \( 1 - 9.39T + 89T^{2} \) |
| 97 | \( 1 + 6.60T + 97T^{2} \) |
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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
−7.75501889325754111727517524387, −6.74341279317117237298326153137, −6.06998238995105563828422665575, −5.25954764209369032842707724133, −4.63703723923924141134175299262, −3.95901680618484646956023379038, −3.18661695878075570850655287260, −2.28394556859152745512399576843, −1.35573493173935979469888687490, 0,
1.35573493173935979469888687490, 2.28394556859152745512399576843, 3.18661695878075570850655287260, 3.95901680618484646956023379038, 4.63703723923924141134175299262, 5.25954764209369032842707724133, 6.06998238995105563828422665575, 6.74341279317117237298326153137, 7.75501889325754111727517524387
