L(s) = 1 | + 2.61·2-s + 3-s + 4.81·4-s + 5-s + 2.61·6-s − 4.73·7-s + 7.35·8-s + 9-s + 2.61·10-s − 3.07·11-s + 4.81·12-s − 6.05·13-s − 12.3·14-s + 15-s + 9.57·16-s − 6.93·17-s + 2.61·18-s − 1.19·19-s + 4.81·20-s − 4.73·21-s − 8.02·22-s + 7.35·24-s + 25-s − 15.8·26-s + 27-s − 22.8·28-s − 4.71·29-s + ⋯ |
L(s) = 1 | + 1.84·2-s + 0.577·3-s + 2.40·4-s + 0.447·5-s + 1.06·6-s − 1.79·7-s + 2.60·8-s + 0.333·9-s + 0.825·10-s − 0.927·11-s + 1.39·12-s − 1.68·13-s − 3.30·14-s + 0.258·15-s + 2.39·16-s − 1.68·17-s + 0.615·18-s − 0.273·19-s + 1.07·20-s − 1.03·21-s − 1.71·22-s + 1.50·24-s + 0.200·25-s − 3.10·26-s + 0.192·27-s − 4.31·28-s − 0.874·29-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 | \( 1 - 2.61T + 2T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 + 3.07T + 11T^{2} \) |
| 13 | \( 1 + 6.05T + 13T^{2} \) |
| 17 | \( 1 + 6.93T + 17T^{2} \) |
| 19 | \( 1 + 1.19T + 19T^{2} \) |
| 29 | \( 1 + 4.71T + 29T^{2} \) |
| 31 | \( 1 - 5.07T + 31T^{2} \) |
| 37 | \( 1 - 1.65T + 37T^{2} \) |
| 41 | \( 1 + 3.95T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 + 5.37T + 47T^{2} \) |
| 53 | \( 1 - 6.44T + 53T^{2} \) |
| 59 | \( 1 + 9.32T + 59T^{2} \) |
| 61 | \( 1 - 4.11T + 61T^{2} \) |
| 67 | \( 1 - 3.93T + 67T^{2} \) |
| 71 | \( 1 + 7.33T + 71T^{2} \) |
| 73 | \( 1 - 2.18T + 73T^{2} \) |
| 79 | \( 1 + 3.68T + 79T^{2} \) |
| 83 | \( 1 + 9.41T + 83T^{2} \) |
| 89 | \( 1 + 10.5T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−7.02493122121454869916210689923, −6.77271239765889653068335369862, −5.98712947134656052391114161243, −5.36700312067396081423933116994, −4.52960945611243142901375036476, −4.02455123585376032742415586888, −2.94133411867608855498340000783, −2.67746023309813671170688738223, −2.07727689146629686380693618601, 0,
2.07727689146629686380693618601, 2.67746023309813671170688738223, 2.94133411867608855498340000783, 4.02455123585376032742415586888, 4.52960945611243142901375036476, 5.36700312067396081423933116994, 5.98712947134656052391114161243, 6.77271239765889653068335369862, 7.02493122121454869916210689923