| L(s) = 1 | + 5-s − 2.50·7-s − 3.27·11-s + 13-s − 4.50·17-s + 3.23·19-s + 2.50·23-s + 25-s + 7.77·29-s − 5.00·31-s − 2.50·35-s + 7.73·37-s − 7.73·41-s + 4·43-s − 9.00·47-s − 0.733·49-s − 6.27·53-s − 3.27·55-s + 9.27·61-s + 65-s + 15.5·67-s − 5.73·71-s + 15.7·73-s + 8.19·77-s − 4.72·79-s + 11.5·83-s − 4.50·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.946·7-s − 0.986·11-s + 0.277·13-s − 1.09·17-s + 0.741·19-s + 0.521·23-s + 0.200·25-s + 1.44·29-s − 0.899·31-s − 0.423·35-s + 1.27·37-s − 1.20·41-s + 0.609·43-s − 1.31·47-s − 0.104·49-s − 0.862·53-s − 0.441·55-s + 1.18·61-s + 0.124·65-s + 1.90·67-s − 0.680·71-s + 1.84·73-s + 0.933·77-s − 0.531·79-s + 1.26·83-s − 0.488·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.530172889\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.530172889\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2.50T + 7T^{2} \) |
| 11 | \( 1 + 3.27T + 11T^{2} \) |
| 17 | \( 1 + 4.50T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 23 | \( 1 - 2.50T + 23T^{2} \) |
| 29 | \( 1 - 7.77T + 29T^{2} \) |
| 31 | \( 1 + 5.00T + 31T^{2} \) |
| 37 | \( 1 - 7.73T + 37T^{2} \) |
| 41 | \( 1 + 7.73T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 9.00T + 47T^{2} \) |
| 53 | \( 1 + 6.27T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 9.27T + 61T^{2} \) |
| 67 | \( 1 - 15.5T + 67T^{2} \) |
| 71 | \( 1 + 5.73T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 + 4.72T + 79T^{2} \) |
| 83 | \( 1 - 11.5T + 83T^{2} \) |
| 89 | \( 1 - 15.2T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−8.326513032378323745382668058545, −7.58737837036573858090325605707, −6.62895053304988908201674097988, −6.34418211677649098240280757299, −5.29326176529849271978199074445, −4.77027287288402940525542307607, −3.59536879944592642442972407312, −2.89076283881792973530101427080, −2.06490471596421162345683651564, −0.66421042371712788949939497322,
0.66421042371712788949939497322, 2.06490471596421162345683651564, 2.89076283881792973530101427080, 3.59536879944592642442972407312, 4.77027287288402940525542307607, 5.29326176529849271978199074445, 6.34418211677649098240280757299, 6.62895053304988908201674097988, 7.58737837036573858090325605707, 8.326513032378323745382668058545
