| L(s) = 1 | + 2-s − 2.48·3-s + 4-s + 1.32·5-s − 2.48·6-s + 7-s + 8-s + 3.18·9-s + 1.32·10-s + 1.36·11-s − 2.48·12-s + 0.281·13-s + 14-s − 3.30·15-s + 16-s − 0.670·17-s + 3.18·18-s − 2.87·19-s + 1.32·20-s − 2.48·21-s + 1.36·22-s − 1.59·23-s − 2.48·24-s − 3.23·25-s + 0.281·26-s − 0.466·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.43·3-s + 0.5·4-s + 0.593·5-s − 1.01·6-s + 0.377·7-s + 0.353·8-s + 1.06·9-s + 0.419·10-s + 0.411·11-s − 0.718·12-s + 0.0781·13-s + 0.267·14-s − 0.852·15-s + 0.250·16-s − 0.162·17-s + 0.751·18-s − 0.659·19-s + 0.296·20-s − 0.542·21-s + 0.290·22-s − 0.333·23-s − 0.507·24-s − 0.647·25-s + 0.0552·26-s − 0.0898·27-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 431 | \( 1 + T \) |
| good | 3 | \( 1 + 2.48T + 3T^{2} \) |
| 5 | \( 1 - 1.32T + 5T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 - 0.281T + 13T^{2} \) |
| 17 | \( 1 + 0.670T + 17T^{2} \) |
| 19 | \( 1 + 2.87T + 19T^{2} \) |
| 23 | \( 1 + 1.59T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 + 1.39T + 31T^{2} \) |
| 37 | \( 1 + 9.31T + 37T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 43 | \( 1 + 12.7T + 43T^{2} \) |
| 47 | \( 1 - 8.45T + 47T^{2} \) |
| 53 | \( 1 + 3.58T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 + 2.22T + 61T^{2} \) |
| 67 | \( 1 + 2.81T + 67T^{2} \) |
| 71 | \( 1 - 11.6T + 71T^{2} \) |
| 73 | \( 1 - 2.26T + 73T^{2} \) |
| 79 | \( 1 + 7.76T + 79T^{2} \) |
| 83 | \( 1 + 0.662T + 83T^{2} \) |
| 89 | \( 1 + 2.92T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
| 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
−7.41678230999421758453567877758, −6.58164577440872504893329441670, −6.30762174602575266337879028156, −5.42727618621405771024243841353, −5.10928445008737399145552923449, −4.25247664643320960138723295203, −3.46320840338049970092230562163, −2.14177543396146075064303146585, −1.43451868811917096766910734942, 0,
1.43451868811917096766910734942, 2.14177543396146075064303146585, 3.46320840338049970092230562163, 4.25247664643320960138723295203, 5.10928445008737399145552923449, 5.42727618621405771024243841353, 6.30762174602575266337879028156, 6.58164577440872504893329441670, 7.41678230999421758453567877758
