| L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s + 3·8-s + 9-s + 10-s − 4·11-s + 12-s − 2·13-s + 15-s − 16-s − 2·17-s − 18-s + 20-s + 4·22-s − 3·24-s + 25-s + 2·26-s − 27-s + 6·29-s − 30-s + 8·31-s − 5·32-s + 4·33-s + 2·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.06·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s − 0.554·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.223·20-s + 0.852·22-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 1.43·31-s − 0.883·32-s + 0.696·33-s + 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.80277751073300, −12.16151453333311, −11.73256028784443, −11.32035075424859, −10.63852587746390, −10.38456466777120, −10.12858738211675, −9.593023856369489, −9.032773212035979, −8.513670327917224, −8.166126088443411, −7.838929723138474, −7.154303779548776, −6.906648540247058, −6.332709247944025, −5.562526462668677, −5.169852644225461, −4.798293539325002, −4.326841954326376, −3.824251362289761, −3.064032213535053, −2.526646353075593, −1.918534209621405, −1.114404077579034, −0.5567899278358745, 0,
0.5567899278358745, 1.114404077579034, 1.918534209621405, 2.526646353075593, 3.064032213535053, 3.824251362289761, 4.326841954326376, 4.798293539325002, 5.169852644225461, 5.562526462668677, 6.332709247944025, 6.906648540247058, 7.154303779548776, 7.838929723138474, 8.166126088443411, 8.513670327917224, 9.032773212035979, 9.593023856369489, 10.12858738211675, 10.38456466777120, 10.63852587746390, 11.32035075424859, 11.73256028784443, 12.16151453333311, 12.80277751073300
