| L(s) = 1 | − 2-s − 3.20·3-s + 4-s − 3.26·5-s + 3.20·6-s + 7-s − 8-s + 7.30·9-s + 3.26·10-s + 1.75·11-s − 3.20·12-s − 5.78·13-s − 14-s + 10.4·15-s + 16-s − 4.31·17-s − 7.30·18-s − 1.34·19-s − 3.26·20-s − 3.20·21-s − 1.75·22-s − 3.06·23-s + 3.20·24-s + 5.64·25-s + 5.78·26-s − 13.8·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.85·3-s + 0.5·4-s − 1.45·5-s + 1.31·6-s + 0.377·7-s − 0.353·8-s + 2.43·9-s + 1.03·10-s + 0.528·11-s − 0.926·12-s − 1.60·13-s − 0.267·14-s + 2.70·15-s + 0.250·16-s − 1.04·17-s − 1.72·18-s − 0.308·19-s − 0.729·20-s − 0.700·21-s − 0.373·22-s − 0.639·23-s + 0.655·24-s + 1.12·25-s + 1.13·26-s − 2.65·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1268770648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1268770648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 3.20T + 3T^{2} \) |
| 5 | \( 1 + 3.26T + 5T^{2} \) |
| 11 | \( 1 - 1.75T + 11T^{2} \) |
| 13 | \( 1 + 5.78T + 13T^{2} \) |
| 17 | \( 1 + 4.31T + 17T^{2} \) |
| 19 | \( 1 + 1.34T + 19T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 - 4.81T + 29T^{2} \) |
| 31 | \( 1 - 2.52T + 31T^{2} \) |
| 37 | \( 1 + 1.68T + 37T^{2} \) |
| 41 | \( 1 - 9.19T + 41T^{2} \) |
| 43 | \( 1 + 4.91T + 43T^{2} \) |
| 47 | \( 1 + 0.249T + 47T^{2} \) |
| 53 | \( 1 - 5.24T + 53T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 + 8.52T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 9.97T + 71T^{2} \) |
| 73 | \( 1 - 3.94T + 73T^{2} \) |
| 79 | \( 1 - 7.76T + 79T^{2} \) |
| 83 | \( 1 + 2.26T + 83T^{2} \) |
| 89 | \( 1 + 11.5T + 89T^{2} \) |
| 97 | \( 1 + 16.2T + 97T^{2} \) |
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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
−7.83736354802432311618254243941, −7.33102515253738392313681291109, −6.75949329041306802694558720700, −6.13980347646498002518021333055, −5.14735405738047545689540289119, −4.47042482120645643466486520473, −4.07633234437312940335454347785, −2.59026854863093639496251194374, −1.35553439230939630554984502141, −0.24681760239353689951403627177,
0.24681760239353689951403627177, 1.35553439230939630554984502141, 2.59026854863093639496251194374, 4.07633234437312940335454347785, 4.47042482120645643466486520473, 5.14735405738047545689540289119, 6.13980347646498002518021333055, 6.75949329041306802694558720700, 7.33102515253738392313681291109, 7.83736354802432311618254243941
