| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s − 12-s − 2·13-s − 14-s + 15-s + 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 21-s − 4·23-s + 24-s + 25-s + 2·26-s − 27-s + 28-s + 6·29-s − 30-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 − 0.288·12-s − 0.554·13-s − 0.267·14-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 − 0.218·21-s − 0.834·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25410 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−15.62056090401213, −15.36777515750124, −14.52183069905609, −14.26979201044466, −13.30877889974693, −12.83520027694701, −12.20470362867467, −11.65815459934789, −11.40309694404146, −10.67546592195713, −10.11284869338216, −9.851837551366737, −8.885142202098090, −8.380489765191005, −8.039476380269426, −7.175796385926023, −6.775539641253657, −6.211686791401691, −5.414578846604732, −4.774577776038766, −4.189709327335086, −3.414101020089112, −2.417436002601086, −1.862251360734791, −0.8224243289739025, 0,
0.8224243289739025, 1.862251360734791, 2.417436002601086, 3.414101020089112, 4.189709327335086, 4.774577776038766, 5.414578846604732, 6.211686791401691, 6.775539641253657, 7.175796385926023, 8.039476380269426, 8.380489765191005, 8.885142202098090, 9.851837551366737, 10.11284869338216, 10.67546592195713, 11.40309694404146, 11.65815459934789, 12.20470362867467, 12.83520027694701, 13.30877889974693, 14.26979201044466, 14.52183069905609, 15.36777515750124, 15.62056090401213
