| L(s) = 1 | − 1.53·2-s − 3-s + 0.343·4-s + 5-s + 1.53·6-s + 2.53·8-s + 9-s − 1.53·10-s − 11-s − 0.343·12-s − 5.35·13-s − 15-s − 4.56·16-s + 1.31·17-s − 1.53·18-s − 5.88·19-s + 0.343·20-s + 1.53·22-s − 0.921·23-s − 2.53·24-s + 25-s + 8.20·26-s − 27-s + 8.16·29-s + 1.53·30-s − 3.50·31-s + 1.92·32-s + ⋯ |
| L(s) = 1 | − 1.08·2-s − 0.577·3-s + 0.171·4-s + 0.447·5-s + 0.624·6-s + 0.896·8-s + 0.333·9-s − 0.484·10-s − 0.301·11-s − 0.0990·12-s − 1.48·13-s − 0.258·15-s − 1.14·16-s + 0.318·17-s − 0.360·18-s − 1.35·19-s + 0.0767·20-s + 0.326·22-s − 0.192·23-s − 0.517·24-s + 0.200·25-s + 1.60·26-s − 0.192·27-s + 1.51·29-s + 0.279·30-s − 0.629·31-s + 0.339·32-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 | \( 1 + 1.53T + 2T^{2} \) |
| 13 | \( 1 + 5.35T + 13T^{2} \) |
| 17 | \( 1 - 1.31T + 17T^{2} \) |
| 19 | \( 1 + 5.88T + 19T^{2} \) |
| 23 | \( 1 + 0.921T + 23T^{2} \) |
| 29 | \( 1 - 8.16T + 29T^{2} \) |
| 31 | \( 1 + 3.50T + 31T^{2} \) |
| 37 | \( 1 - 5.21T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 1.17T + 43T^{2} \) |
| 47 | \( 1 + 1.95T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 10.6T + 59T^{2} \) |
| 61 | \( 1 - 0.173T + 61T^{2} \) |
| 67 | \( 1 + 5.10T + 67T^{2} \) |
| 71 | \( 1 + 9.02T + 71T^{2} \) |
| 73 | \( 1 - 0.385T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 1.73T + 89T^{2} \) |
| 97 | \( 1 - 0.868T + 97T^{2} \) |
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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
−7.65283648805034201538286691344, −6.91358428871896334989789556552, −6.23643559765957268573661208424, −5.41961577105297310376850621649, −4.66307256105993771062325326647, −4.19136528659238139717879924580, −2.70982713604386833313294056184, −2.05719380374159539437357443019, −0.956828274105606286026026009576, 0,
0.956828274105606286026026009576, 2.05719380374159539437357443019, 2.70982713604386833313294056184, 4.19136528659238139717879924580, 4.66307256105993771062325326647, 5.41961577105297310376850621649, 6.23643559765957268573661208424, 6.91358428871896334989789556552, 7.65283648805034201538286691344
