| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 11-s + 13-s + 14-s + 16-s − 17-s − 19-s + 22-s + 23-s + 26-s + 28-s + 29-s + 31-s + 32-s − 34-s − 37-s − 38-s − 43-s + 44-s + 46-s − 47-s + 49-s + 52-s − 53-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 11-s + 13-s + 14-s + 16-s − 17-s − 19-s + 22-s + 23-s + 26-s + 28-s + 29-s + 31-s + 32-s − 34-s − 37-s − 38-s − 43-s + 44-s + 46-s − 47-s + 49-s + 52-s − 53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 615 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 615 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(5.619269431\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.619269431\) |
| \(L(1)\) |
\(\approx\) |
\(2.533624852\) |
| \(L(1)\) |
\(\approx\) |
\(2.533624852\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 41 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−22.9627144167636503580645466835, −21.92113818107795163130253471334, −21.2643754268928962797330323191, −20.59413305847937764974932264620, −19.72769620136849630979295298997, −18.88040953728257141726088211663, −17.58763812297515179692909755442, −17.03454257684956110220989531196, −15.86869911418098927554072549286, −15.16170839650892102872040617298, −14.36831338409247584226633899767, −13.65578986617786536566740442331, −12.79979123421597211007991673260, −11.7348110520082273485832271024, −11.19344066339408606433914522204, −10.384106536231546951107814700279, −8.84802356120396068534814696483, −8.14844615699236324786648536713, −6.78181727845435514348532504866, −6.314791415131720872800289180369, −4.985766782935828270759130967467, −4.357216259352329297097837985664, −3.35372609427092611083212962605, −2.0659711880217313023649907774, −1.17513953369242433096991836101,
1.17513953369242433096991836101, 2.0659711880217313023649907774, 3.35372609427092611083212962605, 4.357216259352329297097837985664, 4.985766782935828270759130967467, 6.314791415131720872800289180369, 6.78181727845435514348532504866, 8.14844615699236324786648536713, 8.84802356120396068534814696483, 10.384106536231546951107814700279, 11.19344066339408606433914522204, 11.7348110520082273485832271024, 12.79979123421597211007991673260, 13.65578986617786536566740442331, 14.36831338409247584226633899767, 15.16170839650892102872040617298, 15.86869911418098927554072549286, 17.03454257684956110220989531196, 17.58763812297515179692909755442, 18.88040953728257141726088211663, 19.72769620136849630979295298997, 20.59413305847937764974932264620, 21.2643754268928962797330323191, 21.92113818107795163130253471334, 22.9627144167636503580645466835
