| L(s) = 1 | − 2-s + 3-s + 4-s + 1.40·5-s − 6-s − 0.998·7-s − 8-s + 9-s − 1.40·10-s + 4.83·11-s + 12-s + 13-s + 0.998·14-s + 1.40·15-s + 16-s − 2.95·17-s − 18-s + 1.88·19-s + 1.40·20-s − 0.998·21-s − 4.83·22-s − 2.84·23-s − 24-s − 3.01·25-s − 26-s + 27-s − 0.998·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.629·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.445·10-s + 1.45·11-s + 0.288·12-s + 0.277·13-s + 0.266·14-s + 0.363·15-s + 0.250·16-s − 0.716·17-s − 0.235·18-s + 0.433·19-s + 0.314·20-s − 0.217·21-s − 1.02·22-s − 0.593·23-s − 0.204·24-s − 0.603·25-s − 0.196·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 5 | \( 1 - 1.40T + 5T^{2} \) |
| 7 | \( 1 + 0.998T + 7T^{2} \) |
| 11 | \( 1 - 4.83T + 11T^{2} \) |
| 17 | \( 1 + 2.95T + 17T^{2} \) |
| 19 | \( 1 - 1.88T + 19T^{2} \) |
| 23 | \( 1 + 2.84T + 23T^{2} \) |
| 29 | \( 1 + 5.72T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 + 4.75T + 37T^{2} \) |
| 41 | \( 1 + 4.77T + 41T^{2} \) |
| 43 | \( 1 - 9.75T + 43T^{2} \) |
| 47 | \( 1 + 2.46T + 47T^{2} \) |
| 53 | \( 1 + 8.38T + 53T^{2} \) |
| 59 | \( 1 + 8.26T + 59T^{2} \) |
| 61 | \( 1 + 6.11T + 61T^{2} \) |
| 67 | \( 1 + 4.55T + 67T^{2} \) |
| 71 | \( 1 + 3.66T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 + 11.1T + 79T^{2} \) |
| 83 | \( 1 - 1.76T + 83T^{2} \) |
| 89 | \( 1 + 0.860T + 89T^{2} \) |
| 97 | \( 1 + 5.09T + 97T^{2} \) |
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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
−7.41984117638591593661979251013, −6.99943496764360535406612419085, −6.11445188964615441673884550635, −5.76830828101499172889131550931, −4.50377471473726538466494835070, −3.71254303226647613011329268814, −3.07910291002251247772113978990, −1.82916658007768075344499662719, −1.60374082069831512361915448633, 0,
1.60374082069831512361915448633, 1.82916658007768075344499662719, 3.07910291002251247772113978990, 3.71254303226647613011329268814, 4.50377471473726538466494835070, 5.76830828101499172889131550931, 6.11445188964615441673884550635, 6.99943496764360535406612419085, 7.41984117638591593661979251013
