| L(s) = 1 | − 5-s − 3.55·7-s + 2.79·11-s + 3.73·13-s + 7.83·17-s + 6.27·19-s + 23-s + 25-s − 2.75·29-s − 2.48·31-s + 3.55·35-s − 1.55·37-s − 5.78·41-s − 4.54·43-s + 6.27·47-s + 5.61·49-s − 8.89·53-s − 2.79·55-s − 3.31·59-s + 11.2·61-s − 3.73·65-s + 5.52·67-s − 6.34·71-s + 15.3·73-s − 9.91·77-s − 9.62·79-s − 5.83·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.34·7-s + 0.842·11-s + 1.03·13-s + 1.89·17-s + 1.44·19-s + 0.208·23-s + 0.200·25-s − 0.512·29-s − 0.447·31-s + 0.600·35-s − 0.255·37-s − 0.904·41-s − 0.693·43-s + 0.915·47-s + 0.801·49-s − 1.22·53-s − 0.376·55-s − 0.431·59-s + 1.43·61-s − 0.462·65-s + 0.674·67-s − 0.752·71-s + 1.80·73-s − 1.13·77-s − 1.08·79-s − 0.640·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.872026300\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.872026300\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 7 | \( 1 + 3.55T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 - 3.73T + 13T^{2} \) |
| 17 | \( 1 - 7.83T + 17T^{2} \) |
| 19 | \( 1 - 6.27T + 19T^{2} \) |
| 29 | \( 1 + 2.75T + 29T^{2} \) |
| 31 | \( 1 + 2.48T + 31T^{2} \) |
| 37 | \( 1 + 1.55T + 37T^{2} \) |
| 41 | \( 1 + 5.78T + 41T^{2} \) |
| 43 | \( 1 + 4.54T + 43T^{2} \) |
| 47 | \( 1 - 6.27T + 47T^{2} \) |
| 53 | \( 1 + 8.89T + 53T^{2} \) |
| 59 | \( 1 + 3.31T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 5.52T + 67T^{2} \) |
| 71 | \( 1 + 6.34T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 + 9.62T + 79T^{2} \) |
| 83 | \( 1 + 5.83T + 83T^{2} \) |
| 89 | \( 1 + 0.390T + 89T^{2} \) |
| 97 | \( 1 - 0.961T + 97T^{2} \) |
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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
−7.73101191830610626860239319492, −7.08991637496526465922986063239, −6.47741999848811681443634605144, −5.74941702805649743865863114056, −5.19507402007869844321271321894, −3.94977482558191168904860297231, −3.42308137265890646370267167357, −3.08949296168369230736063862162, −1.54412907965128953180658996791, −0.72064640569503056221656053999,
0.72064640569503056221656053999, 1.54412907965128953180658996791, 3.08949296168369230736063862162, 3.42308137265890646370267167357, 3.94977482558191168904860297231, 5.19507402007869844321271321894, 5.74941702805649743865863114056, 6.47741999848811681443634605144, 7.08991637496526465922986063239, 7.73101191830610626860239319492
