| L(s) = 1 | + 0.724·2-s − 3-s − 1.47·4-s + 5-s − 0.724·6-s − 2.51·8-s + 9-s + 0.724·10-s + 11-s + 1.47·12-s + 4.61·13-s − 15-s + 1.12·16-s + 7.19·17-s + 0.724·18-s + 4.67·19-s − 1.47·20-s + 0.724·22-s + 7.38·23-s + 2.51·24-s + 25-s + 3.34·26-s − 27-s − 7.68·29-s − 0.724·30-s − 2.69·31-s + 5.85·32-s + ⋯ |
| L(s) = 1 | + 0.512·2-s − 0.577·3-s − 0.737·4-s + 0.447·5-s − 0.295·6-s − 0.890·8-s + 0.333·9-s + 0.229·10-s + 0.301·11-s + 0.425·12-s + 1.28·13-s − 0.258·15-s + 0.280·16-s + 1.74·17-s + 0.170·18-s + 1.07·19-s − 0.329·20-s + 0.154·22-s + 1.53·23-s + 0.514·24-s + 0.200·25-s + 0.656·26-s − 0.192·27-s − 1.42·29-s − 0.132·30-s − 0.483·31-s + 1.03·32-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
| \(L(1)\) |
\(\approx\) |
\(2.375357043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.375357043\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.724T + 2T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 - 7.19T + 17T^{2} \) |
| 19 | \( 1 - 4.67T + 19T^{2} \) |
| 23 | \( 1 - 7.38T + 23T^{2} \) |
| 29 | \( 1 + 7.68T + 29T^{2} \) |
| 31 | \( 1 + 2.69T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 + 1.89T + 41T^{2} \) |
| 43 | \( 1 - 4.69T + 43T^{2} \) |
| 47 | \( 1 - 11.6T + 47T^{2} \) |
| 53 | \( 1 + 5.19T + 53T^{2} \) |
| 59 | \( 1 + 9.88T + 59T^{2} \) |
| 61 | \( 1 + 15.4T + 61T^{2} \) |
| 67 | \( 1 - 10.9T + 67T^{2} \) |
| 71 | \( 1 - 3.55T + 71T^{2} \) |
| 73 | \( 1 - 6.21T + 73T^{2} \) |
| 79 | \( 1 + 14.8T + 79T^{2} \) |
| 83 | \( 1 + 12.4T + 83T^{2} \) |
| 89 | \( 1 + 2.74T + 89T^{2} \) |
| 97 | \( 1 - 5.91T + 97T^{2} \) |
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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
−7.69691199586427155235543810239, −7.11360867222520758981351998086, −6.02661398882531930263909073067, −5.72388046151528967450694826898, −5.19732141878207055464209971348, −4.35005808246069415518106602362, −3.50596918637006501650391053208, −3.07294923525322221654770346450, −1.47241895524835962954559064616, −0.825007246749722451329648052971,
0.825007246749722451329648052971, 1.47241895524835962954559064616, 3.07294923525322221654770346450, 3.50596918637006501650391053208, 4.35005808246069415518106602362, 5.19732141878207055464209971348, 5.72388046151528967450694826898, 6.02661398882531930263909073067, 7.11360867222520758981351998086, 7.69691199586427155235543810239
