| L(s) = 1 | − 2-s − 3-s + 4-s − 4.39·5-s + 6-s − 7-s − 8-s + 9-s + 4.39·10-s − 1.12·11-s − 12-s + 14-s + 4.39·15-s + 16-s + 3.15·17-s − 18-s + 8.07·19-s − 4.39·20-s + 21-s + 1.12·22-s − 5.40·23-s + 24-s + 14.3·25-s − 27-s − 28-s − 6.74·29-s − 4.39·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.96·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 1.39·10-s − 0.338·11-s − 0.288·12-s + 0.267·14-s + 1.13·15-s + 0.250·16-s + 0.765·17-s − 0.235·18-s + 1.85·19-s − 0.983·20-s + 0.218·21-s + 0.239·22-s − 1.12·23-s + 0.204·24-s + 2.87·25-s − 0.192·27-s − 0.188·28-s − 1.25·29-s − 0.803·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3357103305\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3357103305\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 + 4.39T + 5T^{2} \) |
| 11 | \( 1 + 1.12T + 11T^{2} \) |
| 17 | \( 1 - 3.15T + 17T^{2} \) |
| 19 | \( 1 - 8.07T + 19T^{2} \) |
| 23 | \( 1 + 5.40T + 23T^{2} \) |
| 29 | \( 1 + 6.74T + 29T^{2} \) |
| 31 | \( 1 + 5.58T + 31T^{2} \) |
| 37 | \( 1 + 7.61T + 37T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 43 | \( 1 - 1.58T + 43T^{2} \) |
| 47 | \( 1 - 7.58T + 47T^{2} \) |
| 53 | \( 1 - 8.44T + 53T^{2} \) |
| 59 | \( 1 + 7.27T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 - 5.27T + 67T^{2} \) |
| 71 | \( 1 + 8.67T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 3.52T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 1.22T + 97T^{2} \) |
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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
−7.80292814008070877384578488342, −7.25698842313580676236922018690, −7.04830429226424371277934774000, −5.71129306494110262583576896994, −5.31759614344402679876108476757, −4.12525866221734340420921570069, −3.61683031488179182240335106652, −2.87775570249097325754712383968, −1.39868061127630717315566168554, −0.35964369105988706221638392950,
0.35964369105988706221638392950, 1.39868061127630717315566168554, 2.87775570249097325754712383968, 3.61683031488179182240335106652, 4.12525866221734340420921570069, 5.31759614344402679876108476757, 5.71129306494110262583576896994, 7.04830429226424371277934774000, 7.25698842313580676236922018690, 7.80292814008070877384578488342
