| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 2·11-s + 12-s + 3·13-s − 14-s + 15-s + 16-s − 2·17-s − 18-s + 19-s + 20-s + 21-s + 2·22-s + 8·23-s − 24-s + 25-s − 3·26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.832·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.485·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.218·21-s + 0.426·22-s + 1.66·23-s − 0.204·24-s + 1/5·25-s − 0.588·26-s + 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 13 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.83121577405986, −14.07596915645410, −13.54441452907195, −13.35171272884276, −12.65208766668613, −12.09031480467625, −11.45237080665135, −10.91858441631470, −10.47406299147989, −10.10756599375158, −9.310392941096838, −8.906856941425189, −8.576110004680457, −7.973220277361217, −7.409555265422383, −6.851976166176649, −6.365646397667336, −5.619474983211032, −5.024782688691927, −4.463918751943944, −3.547858816081124, −2.956709354104225, −2.463865463375233, −1.545600697596958, −1.188180629176832, 0,
1.188180629176832, 1.545600697596958, 2.463865463375233, 2.956709354104225, 3.547858816081124, 4.463918751943944, 5.024782688691927, 5.619474983211032, 6.365646397667336, 6.851976166176649, 7.409555265422383, 7.973220277361217, 8.576110004680457, 8.906856941425189, 9.310392941096838, 10.10756599375158, 10.47406299147989, 10.91858441631470, 11.45237080665135, 12.09031480467625, 12.65208766668613, 13.35171272884276, 13.54441452907195, 14.07596915645410, 14.83121577405986
