| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 8-s + 9-s − 2·10-s + 4·11-s − 12-s − 2·13-s − 2·15-s + 16-s − 2·17-s − 18-s + 8·19-s + 2·20-s − 4·22-s + 24-s − 25-s + 2·26-s − 27-s + 2·29-s + 2·30-s − 8·31-s − 32-s − 4·33-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 0.516·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 1.83·19-s + 0.447·20-s − 0.852·22-s + 0.204·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.365·30-s − 1.43·31-s − 0.176·32-s − 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 786 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.149014222\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.149014222\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 131 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.04730012038987893027319123152, −9.535725038220235782593235563575, −8.901349969419320661585586227582, −7.57048781949609640088797846174, −6.87281957943411788418329871720, −5.96170612245609191075999506947, −5.20010436534133050839543742003, −3.80036848047385624886238531457, −2.28826601894317808861932442048, −1.07423692191605700915458139021,
1.07423692191605700915458139021, 2.28826601894317808861932442048, 3.80036848047385624886238531457, 5.20010436534133050839543742003, 5.96170612245609191075999506947, 6.87281957943411788418329871720, 7.57048781949609640088797846174, 8.901349969419320661585586227582, 9.535725038220235782593235563575, 10.04730012038987893027319123152
