| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 4·7-s + 8-s + 9-s − 11-s − 12-s − 13-s + 4·14-s + 16-s + 17-s + 18-s + 5·19-s − 4·21-s − 22-s − 3·23-s − 24-s − 26-s − 27-s + 4·28-s − 9·29-s − 10·31-s + 32-s + 33-s + 34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.288·12-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s + 1.14·19-s − 0.872·21-s − 0.213·22-s − 0.625·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s + 0.755·28-s − 1.67·29-s − 1.79·31-s + 0.176·32-s + 0.174·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 364650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.78575483893605, −12.23397526306362, −11.74404249630847, −11.44423063909520, −11.11351922969681, −10.70211971779863, −10.08472492333539, −9.776990939195626, −9.009269057232825, −8.668316808320075, −7.944344061739466, −7.570782601728255, −7.199960019447014, −6.915759522463819, −5.863975782185359, −5.543806264281045, −5.427132828805836, −4.900131404937657, −4.152352366134782, −3.940321992814990, −3.345920831686304, −2.405686711917840, −2.129652306114153, −1.456121803976449, −0.9259898942173788, 0,
0.9259898942173788, 1.456121803976449, 2.129652306114153, 2.405686711917840, 3.345920831686304, 3.940321992814990, 4.152352366134782, 4.900131404937657, 5.427132828805836, 5.543806264281045, 5.863975782185359, 6.915759522463819, 7.199960019447014, 7.570782601728255, 7.944344061739466, 8.668316808320075, 9.009269057232825, 9.776990939195626, 10.08472492333539, 10.70211971779863, 11.11351922969681, 11.44423063909520, 11.74404249630847, 12.23397526306362, 12.78575483893605
