| L(s) = 1 | + 2-s − 3-s + 4-s + 0.113·5-s − 6-s + 2.65·7-s + 8-s + 9-s + 0.113·10-s − 3.17·11-s − 12-s − 13-s + 2.65·14-s − 0.113·15-s + 16-s + 3.51·17-s + 18-s − 2.25·19-s + 0.113·20-s − 2.65·21-s − 3.17·22-s − 4.90·23-s − 24-s − 4.98·25-s − 26-s − 27-s + 2.65·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.0509·5-s − 0.408·6-s + 1.00·7-s + 0.353·8-s + 0.333·9-s + 0.0360·10-s − 0.956·11-s − 0.288·12-s − 0.277·13-s + 0.708·14-s − 0.0294·15-s + 0.250·16-s + 0.851·17-s + 0.235·18-s − 0.517·19-s + 0.0254·20-s − 0.578·21-s − 0.676·22-s − 1.02·23-s − 0.204·24-s − 0.997·25-s − 0.196·26-s − 0.192·27-s + 0.501·28-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 | \( 1 - 0.113T + 5T^{2} \) |
| 7 | \( 1 - 2.65T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 17 | \( 1 - 3.51T + 17T^{2} \) |
| 19 | \( 1 + 2.25T + 19T^{2} \) |
| 23 | \( 1 + 4.90T + 23T^{2} \) |
| 29 | \( 1 - 0.991T + 29T^{2} \) |
| 31 | \( 1 - 5.57T + 31T^{2} \) |
| 37 | \( 1 + 7.31T + 37T^{2} \) |
| 41 | \( 1 + 8.22T + 41T^{2} \) |
| 43 | \( 1 + 6.12T + 43T^{2} \) |
| 47 | \( 1 + 0.711T + 47T^{2} \) |
| 53 | \( 1 - 7.20T + 53T^{2} \) |
| 59 | \( 1 + 8.39T + 59T^{2} \) |
| 61 | \( 1 + 1.34T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 - 13.6T + 71T^{2} \) |
| 73 | \( 1 + 1.59T + 73T^{2} \) |
| 79 | \( 1 - 5.88T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 + 8.95T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−7.47802136690125144545300962203, −6.66108360753787507032049171819, −5.92209363481544111240129610984, −5.28901310319089163559724981820, −4.83266641285282756502485772184, −4.08420857597503320944079744380, −3.19422871640824781044261541247, −2.17974634778983866313494296360, −1.48330135607175936103693692391, 0,
1.48330135607175936103693692391, 2.17974634778983866313494296360, 3.19422871640824781044261541247, 4.08420857597503320944079744380, 4.83266641285282756502485772184, 5.28901310319089163559724981820, 5.92209363481544111240129610984, 6.66108360753787507032049171819, 7.47802136690125144545300962203
