| L(s) = 1 | − 2-s + 1.12·3-s + 4-s − 0.689·5-s − 1.12·6-s − 3.53·7-s − 8-s − 1.72·9-s + 0.689·10-s + 4.42·11-s + 1.12·12-s + 4.38·13-s + 3.53·14-s − 0.778·15-s + 16-s + 5.89·17-s + 1.72·18-s − 0.381·19-s − 0.689·20-s − 3.98·21-s − 4.42·22-s − 1.47·23-s − 1.12·24-s − 4.52·25-s − 4.38·26-s − 5.33·27-s − 3.53·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.651·3-s + 0.5·4-s − 0.308·5-s − 0.460·6-s − 1.33·7-s − 0.353·8-s − 0.576·9-s + 0.218·10-s + 1.33·11-s + 0.325·12-s + 1.21·13-s + 0.944·14-s − 0.200·15-s + 0.250·16-s + 1.42·17-s + 0.407·18-s − 0.0874·19-s − 0.154·20-s − 0.869·21-s − 0.943·22-s − 0.306·23-s − 0.230·24-s − 0.904·25-s − 0.860·26-s − 1.02·27-s − 0.667·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.421948783\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.421948783\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 1.12T + 3T^{2} \) |
| 5 | \( 1 + 0.689T + 5T^{2} \) |
| 7 | \( 1 + 3.53T + 7T^{2} \) |
| 11 | \( 1 - 4.42T + 11T^{2} \) |
| 13 | \( 1 - 4.38T + 13T^{2} \) |
| 17 | \( 1 - 5.89T + 17T^{2} \) |
| 19 | \( 1 + 0.381T + 19T^{2} \) |
| 23 | \( 1 + 1.47T + 23T^{2} \) |
| 29 | \( 1 - 2.39T + 29T^{2} \) |
| 37 | \( 1 + 7.26T + 37T^{2} \) |
| 41 | \( 1 - 10.7T + 41T^{2} \) |
| 43 | \( 1 + 5.49T + 43T^{2} \) |
| 47 | \( 1 + 5.72T + 47T^{2} \) |
| 53 | \( 1 - 9.81T + 53T^{2} \) |
| 59 | \( 1 - 11.9T + 59T^{2} \) |
| 61 | \( 1 - 2.10T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 + 16.6T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 - 8.32T + 79T^{2} \) |
| 83 | \( 1 - 8.22T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
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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
−8.221985502182459724538822025605, −7.52801639886526889367824393620, −6.68404205045638408474883399355, −6.14044000639208169880847273971, −5.54115001909103708995457963885, −3.89357297507474901345942130262, −3.59837996086649338532810320137, −2.90283274147962970216226374780, −1.74529204733546753501243971262, −0.68138256759540872402112362184,
0.68138256759540872402112362184, 1.74529204733546753501243971262, 2.90283274147962970216226374780, 3.59837996086649338532810320137, 3.89357297507474901345942130262, 5.54115001909103708995457963885, 6.14044000639208169880847273971, 6.68404205045638408474883399355, 7.52801639886526889367824393620, 8.221985502182459724538822025605
