| L(s) = 1 | − 1.65·2-s − 3-s + 0.726·4-s − 2.92·5-s + 1.65·6-s + 7-s + 2.10·8-s + 9-s + 4.82·10-s + 4.37·11-s − 0.726·12-s − 1.65·14-s + 2.92·15-s − 4.92·16-s + 3.65·17-s − 1.65·18-s + 5.75·19-s − 2.12·20-s − 21-s − 7.22·22-s + 7.02·23-s − 2.10·24-s + 3.55·25-s − 27-s + 0.726·28-s + 1.19·29-s − 4.82·30-s + ⋯ |
| L(s) = 1 | − 1.16·2-s − 0.577·3-s + 0.363·4-s − 1.30·5-s + 0.674·6-s + 0.377·7-s + 0.743·8-s + 0.333·9-s + 1.52·10-s + 1.31·11-s − 0.209·12-s − 0.441·14-s + 0.755·15-s − 1.23·16-s + 0.885·17-s − 0.389·18-s + 1.32·19-s − 0.474·20-s − 0.218·21-s − 1.54·22-s + 1.46·23-s − 0.429·24-s + 0.711·25-s − 0.192·27-s + 0.137·28-s + 0.222·29-s − 0.881·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7968551014\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7968551014\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 5 | \( 1 + 2.92T + 5T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 5.75T + 19T^{2} \) |
| 23 | \( 1 - 7.02T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 - 6.47T + 31T^{2} \) |
| 37 | \( 1 - 2.92T + 37T^{2} \) |
| 41 | \( 1 - 8.60T + 41T^{2} \) |
| 43 | \( 1 + 5.72T + 43T^{2} \) |
| 47 | \( 1 - 9.58T + 47T^{2} \) |
| 53 | \( 1 + 0.302T + 53T^{2} \) |
| 59 | \( 1 + 3.02T + 59T^{2} \) |
| 61 | \( 1 + 0.302T + 61T^{2} \) |
| 67 | \( 1 - 8.70T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 0.932T + 73T^{2} \) |
| 79 | \( 1 + 16.9T + 79T^{2} \) |
| 83 | \( 1 - 1.26T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 11.9T + 97T^{2} \) |
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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
−8.572127575433387821062363908523, −7.80355906863026423764399497407, −7.35042526427130675602877750286, −6.70301662142468319904745153970, −5.56618559605753250548155602929, −4.61869339580710429202409874929, −4.05351641740898799198810226947, −3.02245943782334337486286715960, −1.27680981914522532426862860838, −0.793764586305138567554469106742,
0.793764586305138567554469106742, 1.27680981914522532426862860838, 3.02245943782334337486286715960, 4.05351641740898799198810226947, 4.61869339580710429202409874929, 5.56618559605753250548155602929, 6.70301662142468319904745153970, 7.35042526427130675602877750286, 7.80355906863026423764399497407, 8.572127575433387821062363908523
