| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s − 7-s − 8-s + 9-s − 10-s + 4·11-s + 12-s + 7·13-s + 14-s + 15-s + 16-s − 18-s − 6·19-s + 20-s − 21-s − 4·22-s − 23-s − 24-s + 25-s − 7·26-s + 27-s − 28-s + 3·29-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 + 1.20·11-s + 0.288·12-s + 1.94·13-s + 0.267·14-s + 0.258·15-s + 1/4·16-s − 0.235·18-s − 1.37·19-s + 0.223·20-s − 0.218·21-s − 0.852·22-s − 0.208·23-s − 0.204·24-s + 1/5·25-s − 1.37·26-s + 0.192·27-s − 0.188·28-s + 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 372810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.798006661\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.798006661\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 3 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−12.53507701001627, −12.06128109991533, −11.32749096381973, −11.09201765021470, −10.73526636095811, −10.12091524152810, −9.657182534942531, −9.169901137551096, −9.026580153822662, −8.456883444194442, −7.980127322939091, −7.733880518973151, −6.798181224701409, −6.449877885459101, −6.124220105583067, −5.979238529598494, −4.834731735172570, −4.389768798310745, −3.870455093238819, −3.290445449666139, −2.966825014229275, −2.033139070863154, −1.779093275726874, −1.093945488853996, −0.5942005973462560,
0.5942005973462560, 1.093945488853996, 1.779093275726874, 2.033139070863154, 2.966825014229275, 3.290445449666139, 3.870455093238819, 4.389768798310745, 4.834731735172570, 5.979238529598494, 6.124220105583067, 6.449877885459101, 6.798181224701409, 7.733880518973151, 7.980127322939091, 8.456883444194442, 9.026580153822662, 9.169901137551096, 9.657182534942531, 10.12091524152810, 10.73526636095811, 11.09201765021470, 11.32749096381973, 12.06128109991533, 12.53507701001627
