| L(s) = 1 | + (−0.568 − 0.822i)2-s + (2.85 + 0.704i)3-s + (−0.354 + 0.935i)4-s + (−0.748 + 0.663i)5-s + (−1.04 − 2.75i)6-s + (−1.55 − 0.384i)7-s + (0.970 − 0.239i)8-s + (5.00 + 2.62i)9-s + (0.970 + 0.239i)10-s + (3.44 + 3.05i)11-s + (−1.67 + 2.42i)12-s + (2.30 − 6.07i)13-s + (0.569 + 1.50i)14-s + (−2.60 + 1.36i)15-s + (−0.748 − 0.663i)16-s + (−1.00 + 2.63i)17-s + ⋯ |
| L(s) = 1 | + (−0.401 − 0.581i)2-s + (1.64 + 0.406i)3-s + (−0.177 + 0.467i)4-s + (−0.334 + 0.296i)5-s + (−0.425 − 1.12i)6-s + (−0.589 − 0.145i)7-s + (0.343 − 0.0846i)8-s + (1.66 + 0.875i)9-s + (0.307 + 0.0756i)10-s + (1.03 + 0.920i)11-s + (−0.482 + 0.698i)12-s + (0.639 − 1.68i)13-s + (0.152 + 0.401i)14-s + (−0.672 + 0.352i)15-s + (−0.187 − 0.165i)16-s + (−0.242 + 0.640i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.180i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.07223 + 0.188422i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.07223 + 0.188422i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.568 + 0.822i)T \) |
| 5 | \( 1 + (0.748 - 0.663i)T \) |
| 79 | \( 1 + (3.10 + 8.32i)T \) |
| good | 3 | \( 1 + (-2.85 - 0.704i)T + (2.65 + 1.39i)T^{2} \) |
| 7 | \( 1 + (1.55 + 0.384i)T + (6.19 + 3.25i)T^{2} \) |
| 11 | \( 1 + (-3.44 - 3.05i)T + (1.32 + 10.9i)T^{2} \) |
| 13 | \( 1 + (-2.30 + 6.07i)T + (-9.73 - 8.62i)T^{2} \) |
| 17 | \( 1 + (1.00 - 2.63i)T + (-12.7 - 11.2i)T^{2} \) |
| 19 | \( 1 + (-0.958 - 7.89i)T + (-18.4 + 4.54i)T^{2} \) |
| 23 | \( 1 + 0.497T + 23T^{2} \) |
| 29 | \( 1 + (-1.82 + 0.959i)T + (16.4 - 23.8i)T^{2} \) |
| 31 | \( 1 + (-2.42 - 3.51i)T + (-10.9 + 28.9i)T^{2} \) |
| 37 | \( 1 + (0.849 + 6.99i)T + (-35.9 + 8.85i)T^{2} \) |
| 41 | \( 1 + (-6.23 + 5.52i)T + (4.94 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-2.43 + 2.15i)T + (5.18 - 42.6i)T^{2} \) |
| 47 | \( 1 + (0.923 - 7.60i)T + (-45.6 - 11.2i)T^{2} \) |
| 53 | \( 1 + (2.25 - 0.555i)T + (46.9 - 24.6i)T^{2} \) |
| 59 | \( 1 + (2.25 + 5.94i)T + (-44.1 + 39.1i)T^{2} \) |
| 61 | \( 1 + (-1.71 - 14.1i)T + (-59.2 + 14.5i)T^{2} \) |
| 67 | \( 1 + (6.46 - 9.36i)T + (-23.7 - 62.6i)T^{2} \) |
| 71 | \( 1 + (-1.51 + 0.372i)T + (62.8 - 32.9i)T^{2} \) |
| 73 | \( 1 + (2.48 + 6.53i)T + (-54.6 + 48.4i)T^{2} \) |
| 83 | \( 1 + (-4.02 + 10.6i)T + (-62.1 - 55.0i)T^{2} \) |
| 89 | \( 1 + (-9.75 - 2.40i)T + (78.8 + 41.3i)T^{2} \) |
| 97 | \( 1 + (1.53 + 12.6i)T + (-94.1 + 23.2i)T^{2} \) |
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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
−10.23640731911169094584599528264, −9.479271433826321053695286073706, −8.694152864630507329800247076407, −7.964834308688210614913451033503, −7.33683191550104372939891080047, −6.00759228027453796506812823272, −4.19120193327405452964602972764, −3.64607744244401356201266779395, −2.86609835786157972100394809862, −1.57680031324009768849997635987,
1.17722190630927247655634561469, 2.62314749046327124421871117663, 3.69943980762342101144000627018, 4.64824622022481213749641420415, 6.52616375923941555818529470429, 6.72317327940276331812638438483, 7.88586800303679678641448899710, 8.652227906772792144456700628659, 9.345847967859213717467253117553, 9.395251244369214685028943985215
