| L(s) = 1 | − 5-s + 2.13·7-s + 1.52·11-s − 3.42·13-s + 2.08·17-s + 7.10·19-s + 6.73·23-s + 25-s + 3.83·29-s − 5.67·31-s − 2.13·35-s − 37-s + 0.0328·41-s + 2.26·43-s + 3.10·47-s − 2.44·49-s − 11.6·53-s − 1.52·55-s + 10.6·59-s − 11.4·61-s + 3.42·65-s − 3.95·67-s + 1.66·71-s + 7.60·73-s + 3.25·77-s − 7.30·79-s + 12.2·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.806·7-s + 0.460·11-s − 0.951·13-s + 0.505·17-s + 1.63·19-s + 1.40·23-s + 0.200·25-s + 0.712·29-s − 1.01·31-s − 0.360·35-s − 0.164·37-s + 0.00513·41-s + 0.345·43-s + 0.452·47-s − 0.349·49-s − 1.59·53-s − 0.205·55-s + 1.38·59-s − 1.47·61-s + 0.425·65-s − 0.482·67-s + 0.197·71-s + 0.889·73-s + 0.371·77-s − 0.821·79-s + 1.34·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.192894164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.192894164\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 - 2.13T + 7T^{2} \) |
| 11 | \( 1 - 1.52T + 11T^{2} \) |
| 13 | \( 1 + 3.42T + 13T^{2} \) |
| 17 | \( 1 - 2.08T + 17T^{2} \) |
| 19 | \( 1 - 7.10T + 19T^{2} \) |
| 23 | \( 1 - 6.73T + 23T^{2} \) |
| 29 | \( 1 - 3.83T + 29T^{2} \) |
| 31 | \( 1 + 5.67T + 31T^{2} \) |
| 41 | \( 1 - 0.0328T + 41T^{2} \) |
| 43 | \( 1 - 2.26T + 43T^{2} \) |
| 47 | \( 1 - 3.10T + 47T^{2} \) |
| 53 | \( 1 + 11.6T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 3.95T + 67T^{2} \) |
| 71 | \( 1 - 1.66T + 71T^{2} \) |
| 73 | \( 1 - 7.60T + 73T^{2} \) |
| 79 | \( 1 + 7.30T + 79T^{2} \) |
| 83 | \( 1 - 12.2T + 83T^{2} \) |
| 89 | \( 1 + 9.08T + 89T^{2} \) |
| 97 | \( 1 + 6.76T + 97T^{2} \) |
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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
−7.82098093028301072223322812816, −7.39546375164778094654576516542, −6.80349226744687564912534823620, −5.72368439923866862963170442634, −5.03013587163120395172448067677, −4.57652490156093888241598962579, −3.49356651609409531636791081603, −2.88628939781221405680687186011, −1.69914417260477253635701151949, −0.802681604587713076730340066787,
0.802681604587713076730340066787, 1.69914417260477253635701151949, 2.88628939781221405680687186011, 3.49356651609409531636791081603, 4.57652490156093888241598962579, 5.03013587163120395172448067677, 5.72368439923866862963170442634, 6.80349226744687564912534823620, 7.39546375164778094654576516542, 7.82098093028301072223322812816
