| L(s) = 1 | − 1.70·3-s − 5-s − 0.630·7-s − 0.0783·9-s − 0.290·11-s − 0.921·13-s + 1.70·15-s + 4.97·17-s + 6.04·19-s + 1.07·21-s − 2.29·23-s + 25-s + 5.26·27-s + 29-s − 10.0·31-s + 0.496·33-s + 0.630·35-s + 1.55·37-s + 1.57·39-s + 0.340·41-s + 5.70·43-s + 0.0783·45-s + 1.12·47-s − 6.60·49-s − 8.49·51-s − 0.340·53-s + 0.290·55-s + ⋯ |
| L(s) = 1 | − 0.986·3-s − 0.447·5-s − 0.238·7-s − 0.0261·9-s − 0.0876·11-s − 0.255·13-s + 0.441·15-s + 1.20·17-s + 1.38·19-s + 0.235·21-s − 0.477·23-s + 0.200·25-s + 1.01·27-s + 0.185·29-s − 1.80·31-s + 0.0865·33-s + 0.106·35-s + 0.255·37-s + 0.252·39-s + 0.0531·41-s + 0.870·43-s + 0.0116·45-s + 0.164·47-s − 0.943·49-s − 1.18·51-s − 0.0467·53-s + 0.0392·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 1.70T + 3T^{2} \) |
| 7 | \( 1 + 0.630T + 7T^{2} \) |
| 11 | \( 1 + 0.290T + 11T^{2} \) |
| 13 | \( 1 + 0.921T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 - 6.04T + 19T^{2} \) |
| 23 | \( 1 + 2.29T + 23T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 1.55T + 37T^{2} \) |
| 41 | \( 1 - 0.340T + 41T^{2} \) |
| 43 | \( 1 - 5.70T + 43T^{2} \) |
| 47 | \( 1 - 1.12T + 47T^{2} \) |
| 53 | \( 1 + 0.340T + 53T^{2} \) |
| 59 | \( 1 + 9.75T + 59T^{2} \) |
| 61 | \( 1 - 3.07T + 61T^{2} \) |
| 67 | \( 1 - 5.70T + 67T^{2} \) |
| 71 | \( 1 + 9.07T + 71T^{2} \) |
| 73 | \( 1 + 6.94T + 73T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + 2.78T + 83T^{2} \) |
| 89 | \( 1 - 4.73T + 89T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−8.588653931144457430330936773555, −7.62456300728160561758620876105, −7.18446360201357078092142758448, −6.04341870009173345002313835651, −5.55625160752083595672875090853, −4.79716271025353039136340835981, −3.68478541654179143411143881818, −2.86256311432694956236547411841, −1.26525992624491365667413558009, 0,
1.26525992624491365667413558009, 2.86256311432694956236547411841, 3.68478541654179143411143881818, 4.79716271025353039136340835981, 5.55625160752083595672875090853, 6.04341870009173345002313835651, 7.18446360201357078092142758448, 7.62456300728160561758620876105, 8.588653931144457430330936773555
