| L(s) = 1 | + 1.59·2-s − 3-s + 0.550·4-s − 5-s − 1.59·6-s − 3.15·7-s − 2.31·8-s + 9-s − 1.59·10-s − 3.93·11-s − 0.550·12-s + 5.83·13-s − 5.03·14-s + 15-s − 4.79·16-s − 3.95·17-s + 1.59·18-s − 5.47·19-s − 0.550·20-s + 3.15·21-s − 6.28·22-s + 2.31·24-s + 25-s + 9.31·26-s − 27-s − 1.73·28-s − 6.54·29-s + ⋯ |
| L(s) = 1 | + 1.12·2-s − 0.577·3-s + 0.275·4-s − 0.447·5-s − 0.651·6-s − 1.19·7-s − 0.818·8-s + 0.333·9-s − 0.504·10-s − 1.18·11-s − 0.158·12-s + 1.61·13-s − 1.34·14-s + 0.258·15-s − 1.19·16-s − 0.960·17-s + 0.376·18-s − 1.25·19-s − 0.123·20-s + 0.688·21-s − 1.34·22-s + 0.472·24-s + 0.200·25-s + 1.82·26-s − 0.192·27-s − 0.327·28-s − 1.21·29-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
| \(L(1)\) |
\(\approx\) |
\(0.7071080658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7071080658\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.59T + 2T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 11 | \( 1 + 3.93T + 11T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 19 | \( 1 + 5.47T + 19T^{2} \) |
| 29 | \( 1 + 6.54T + 29T^{2} \) |
| 31 | \( 1 - 5.79T + 31T^{2} \) |
| 37 | \( 1 + 4.28T + 37T^{2} \) |
| 41 | \( 1 - 4.15T + 41T^{2} \) |
| 43 | \( 1 + 3.97T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 4.29T + 53T^{2} \) |
| 59 | \( 1 + 13.8T + 59T^{2} \) |
| 61 | \( 1 + 2.38T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 7.43T + 71T^{2} \) |
| 73 | \( 1 - 4.84T + 73T^{2} \) |
| 79 | \( 1 - 9.59T + 79T^{2} \) |
| 83 | \( 1 + 8.40T + 83T^{2} \) |
| 89 | \( 1 - 9.86T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−7.71040100087426576359341308049, −6.74845623764049721791584911940, −6.18247863955922462988995001208, −5.93312719730681235637128187445, −4.88048572513190688828294842893, −4.40166951716738386724412168763, −3.54750307773284417562410161830, −3.11325141418208868794814638118, −2.01000651617617814160460240710, −0.33910390484505960582154813508,
0.33910390484505960582154813508, 2.01000651617617814160460240710, 3.11325141418208868794814638118, 3.54750307773284417562410161830, 4.40166951716738386724412168763, 4.88048572513190688828294842893, 5.93312719730681235637128187445, 6.18247863955922462988995001208, 6.74845623764049721791584911940, 7.71040100087426576359341308049
