| L(s) = 1 | − 0.414·2-s + 3-s − 1.82·4-s + 5-s − 0.414·6-s + 1.58·8-s + 9-s − 0.414·10-s − 2.82·11-s − 1.82·12-s + 0.828·13-s + 15-s + 3·16-s + 3.65·17-s − 0.414·18-s + 4.82·19-s − 1.82·20-s + 1.17·22-s + 3.65·23-s + 1.58·24-s + 25-s − 0.343·26-s + 27-s + 6·29-s − 0.414·30-s − 10.4·31-s − 4.41·32-s + ⋯ |
| L(s) = 1 | − 0.292·2-s + 0.577·3-s − 0.914·4-s + 0.447·5-s − 0.169·6-s + 0.560·8-s + 0.333·9-s − 0.130·10-s − 0.852·11-s − 0.527·12-s + 0.229·13-s + 0.258·15-s + 0.750·16-s + 0.886·17-s − 0.0976·18-s + 1.10·19-s − 0.408·20-s + 0.249·22-s + 0.762·23-s + 0.323·24-s + 0.200·25-s − 0.0672·26-s + 0.192·27-s + 1.11·29-s − 0.0756·30-s − 1.88·31-s − 0.780·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.444646361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.444646361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 - 0.828T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 4.82T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 10.4T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 0.343T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 8.48T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 2.82T + 71T^{2} \) |
| 73 | \( 1 + 16.8T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 9.65T + 83T^{2} \) |
| 89 | \( 1 - 17.3T + 89T^{2} \) |
| 97 | \( 1 + 10.4T + 97T^{2} \) |
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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
−10.11610419301513353725651858355, −9.496471868067076958816553134998, −8.768389630615146284705769388928, −7.906440075237880429298884113272, −7.22457414454124183248320450083, −5.70381390523176382953592231914, −5.02877706373215574098424228984, −3.80849625379078513226595904461, −2.73605747614670992118759005157, −1.11133814432095859866628281063,
1.11133814432095859866628281063, 2.73605747614670992118759005157, 3.80849625379078513226595904461, 5.02877706373215574098424228984, 5.70381390523176382953592231914, 7.22457414454124183248320450083, 7.906440075237880429298884113272, 8.768389630615146284705769388928, 9.496471868067076958816553134998, 10.11610419301513353725651858355
