| L(s) = 1 | + (0.861 + 0.697i)2-s + (0.923 + 1.46i)3-s + (−0.160 − 0.753i)4-s + (1.74 + 1.39i)5-s + (−0.226 + 1.90i)6-s + (−2.43 + 0.651i)7-s + (1.39 − 2.73i)8-s + (−1.29 + 2.70i)9-s + (0.529 + 2.42i)10-s + (2.42 − 0.255i)11-s + (0.956 − 0.930i)12-s + (−0.0493 − 0.0608i)13-s + (−2.54 − 1.13i)14-s + (−0.435 + 3.84i)15-s + (1.70 − 0.759i)16-s + (−5.10 − 2.60i)17-s + ⋯ |
| L(s) = 1 | + (0.609 + 0.493i)2-s + (0.533 + 0.845i)3-s + (−0.0800 − 0.376i)4-s + (0.780 + 0.624i)5-s + (−0.0925 + 0.778i)6-s + (−0.918 + 0.246i)7-s + (0.493 − 0.967i)8-s + (−0.431 + 0.902i)9-s + (0.167 + 0.766i)10-s + (0.732 − 0.0769i)11-s + (0.275 − 0.268i)12-s + (−0.0136 − 0.0168i)13-s + (−0.681 − 0.303i)14-s + (−0.112 + 0.993i)15-s + (0.426 − 0.189i)16-s + (−1.23 − 0.630i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.369 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.59833 + 1.08422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.59833 + 1.08422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.923 - 1.46i)T \) |
| 5 | \( 1 + (-1.74 - 1.39i)T \) |
| good | 2 | \( 1 + (-0.861 - 0.697i)T + (0.415 + 1.95i)T^{2} \) |
| 7 | \( 1 + (2.43 - 0.651i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-2.42 + 0.255i)T + (10.7 - 2.28i)T^{2} \) |
| 13 | \( 1 + (0.0493 + 0.0608i)T + (-2.70 + 12.7i)T^{2} \) |
| 17 | \( 1 + (5.10 + 2.60i)T + (9.99 + 13.7i)T^{2} \) |
| 19 | \( 1 + (-1.99 + 0.648i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (2.24 + 5.84i)T + (-17.0 + 15.3i)T^{2} \) |
| 29 | \( 1 + (-3.64 - 4.05i)T + (-3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (-1.74 + 1.93i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (1.17 + 7.41i)T + (-35.1 + 11.4i)T^{2} \) |
| 41 | \( 1 + (-2.37 - 0.249i)T + (40.1 + 8.52i)T^{2} \) |
| 43 | \( 1 + (-0.621 - 2.31i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (9.33 + 0.489i)T + (46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (-11.3 + 5.76i)T + (31.1 - 42.8i)T^{2} \) |
| 59 | \( 1 + (-1.02 + 9.70i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.14 - 10.8i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (-4.83 + 0.253i)T + (66.6 - 7.00i)T^{2} \) |
| 71 | \( 1 + (5.52 + 1.79i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (2.05 - 12.9i)T + (-69.4 - 22.5i)T^{2} \) |
| 79 | \( 1 + (5.54 - 4.99i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (0.731 + 1.12i)T + (-33.7 + 75.8i)T^{2} \) |
| 89 | \( 1 + (6.60 - 4.80i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (1.00 - 19.2i)T + (-96.4 - 10.1i)T^{2} \) |
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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
−12.86194273477733383569883959599, −11.25126504238465996149529055419, −10.24213767199012904828552352303, −9.605012826414191614155973398891, −8.819320664507416224374770831581, −6.95802632708002244488235895723, −6.25529806336247756419005840431, −5.12909924490184940484181341391, −3.92928668524258188416711043059, −2.57147593533319211202659008908,
1.76489159383479839078215567277, 3.12099936784397101526245923819, 4.31351946986733611059308642548, 5.92654853397086873518908715511, 6.89675310538929496346897337865, 8.215157506026573291472537013644, 9.054519476800670460954545891532, 10.01732436775192109162245720892, 11.59849555424727848054632721934, 12.27197307525859531725195614744
