| L(s) = 1 | − 0.569·2-s + 3-s − 1.67·4-s − 5-s − 0.569·6-s + 2.09·8-s + 9-s + 0.569·10-s + 11-s − 1.67·12-s + 2.00·13-s − 15-s + 2.15·16-s − 4.76·17-s − 0.569·18-s + 3.13·19-s + 1.67·20-s − 0.569·22-s + 2.31·23-s + 2.09·24-s + 25-s − 1.14·26-s + 27-s − 2.53·29-s + 0.569·30-s − 3.75·31-s − 5.41·32-s + ⋯ |
| L(s) = 1 | − 0.403·2-s + 0.577·3-s − 0.837·4-s − 0.447·5-s − 0.232·6-s + 0.740·8-s + 0.333·9-s + 0.180·10-s + 0.301·11-s − 0.483·12-s + 0.556·13-s − 0.258·15-s + 0.539·16-s − 1.15·17-s − 0.134·18-s + 0.719·19-s + 0.374·20-s − 0.121·22-s + 0.481·23-s + 0.427·24-s + 0.200·25-s − 0.224·26-s + 0.192·27-s − 0.471·29-s + 0.104·30-s − 0.673·31-s − 0.957·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8085 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + 0.569T + 2T^{2} \) |
| 13 | \( 1 - 2.00T + 13T^{2} \) |
| 17 | \( 1 + 4.76T + 17T^{2} \) |
| 19 | \( 1 - 3.13T + 19T^{2} \) |
| 23 | \( 1 - 2.31T + 23T^{2} \) |
| 29 | \( 1 + 2.53T + 29T^{2} \) |
| 31 | \( 1 + 3.75T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 3.68T + 41T^{2} \) |
| 43 | \( 1 + 8.37T + 43T^{2} \) |
| 47 | \( 1 + 4.05T + 47T^{2} \) |
| 53 | \( 1 - 6.96T + 53T^{2} \) |
| 59 | \( 1 + 3.59T + 59T^{2} \) |
| 61 | \( 1 - 10.7T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 - 0.837T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.01T + 79T^{2} \) |
| 83 | \( 1 + 1.20T + 83T^{2} \) |
| 89 | \( 1 - 14.0T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−7.50363831726938507861370149669, −7.10894471043032655732751043553, −6.18061297416609733166307312895, −5.17902442019019296769520653264, −4.63548806864752766962455817725, −3.72951854740891164880574666580, −3.38013751282507790518389738315, −2.08828446934880362684932910887, −1.18266827227271393546940822184, 0,
1.18266827227271393546940822184, 2.08828446934880362684932910887, 3.38013751282507790518389738315, 3.72951854740891164880574666580, 4.63548806864752766962455817725, 5.17902442019019296769520653264, 6.18061297416609733166307312895, 7.10894471043032655732751043553, 7.50363831726938507861370149669
