| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 8-s + 9-s − 10-s − 12-s − 13-s + 15-s + 16-s + 2·17-s + 18-s + 4·19-s − 20-s + 4·23-s − 24-s + 25-s − 26-s − 27-s − 2·29-s + 30-s − 4·31-s + 32-s + 2·34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.258·15-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s + 0.834·23-s − 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s − 0.371·29-s + 0.182·30-s − 0.718·31-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 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
−14.86092561296616, −14.34947882330278, −13.79198263991231, −13.28605974282746, −12.69091142400548, −12.24543245787063, −11.91307242219075, −11.22934807726007, −10.91271112343489, −10.38973539403064, −9.557208724811472, −9.325415640305780, −8.380726470461638, −7.819356442515000, −7.268325283517110, −6.861129068823503, −6.206559131936982, −5.460383738427065, −5.196600828789415, −4.554936841748617, −3.853199779447336, −3.308912187867003, −2.695564913907786, −1.722768932377989, −1.030999446183115, 0,
1.030999446183115, 1.722768932377989, 2.695564913907786, 3.308912187867003, 3.853199779447336, 4.554936841748617, 5.196600828789415, 5.460383738427065, 6.206559131936982, 6.861129068823503, 7.268325283517110, 7.819356442515000, 8.380726470461638, 9.325415640305780, 9.557208724811472, 10.38973539403064, 10.91271112343489, 11.22934807726007, 11.91307242219075, 12.24543245787063, 12.69091142400548, 13.28605974282746, 13.79198263991231, 14.34947882330278, 14.86092561296616
