| L(s) = 1 | − 2-s − 4-s − 4·5-s + 7-s + 3·8-s − 3·9-s + 4·10-s + 4·11-s − 4·13-s − 14-s − 16-s + 3·18-s + 6·19-s + 4·20-s − 4·22-s + 4·23-s + 11·25-s + 4·26-s − 28-s + 6·29-s − 2·31-s − 5·32-s − 4·35-s + 3·36-s − 6·38-s − 12·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 0.377·7-s + 1.06·8-s − 9-s + 1.26·10-s + 1.20·11-s − 1.10·13-s − 0.267·14-s − 1/4·16-s + 0.707·18-s + 1.37·19-s + 0.894·20-s − 0.852·22-s + 0.834·23-s + 11/5·25-s + 0.784·26-s − 0.188·28-s + 1.11·29-s − 0.359·31-s − 0.883·32-s − 0.676·35-s + 1/2·36-s − 0.973·38-s − 1.89·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9583 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9583 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6342698831\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6342698831\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 37 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−7.80581973633186001613085906520, −7.20345317234480834695613940387, −6.74142156044087859809119546072, −5.34047730153668077277125331206, −4.92430723655081644878916410559, −4.14462574956659317826264230734, −3.52805688613658527543113840387, −2.73832625295336789221868833529, −1.26382588370343781872419923367, −0.49071914489100708263185185683,
0.49071914489100708263185185683, 1.26382588370343781872419923367, 2.73832625295336789221868833529, 3.52805688613658527543113840387, 4.14462574956659317826264230734, 4.92430723655081644878916410559, 5.34047730153668077277125331206, 6.74142156044087859809119546072, 7.20345317234480834695613940387, 7.80581973633186001613085906520
