| L(s) = 1 | + (−0.248 − 0.968i)2-s + (2.88 − 0.550i)3-s + (−0.876 + 0.481i)4-s + (2.01 − 0.968i)5-s + (−1.25 − 2.66i)6-s + (−0.797 + 1.09i)7-s + (0.684 + 0.728i)8-s + (5.24 − 2.07i)9-s + (−1.43 − 1.71i)10-s + (−5.82 + 1.49i)11-s + (−2.26 + 1.87i)12-s + (0.617 + 1.55i)13-s + (1.26 + 0.499i)14-s + (5.28 − 3.90i)15-s + (0.535 − 0.844i)16-s + (−1.13 + 2.06i)17-s + ⋯ |
| L(s) = 1 | + (−0.175 − 0.684i)2-s + (1.66 − 0.318i)3-s + (−0.438 + 0.240i)4-s + (0.901 − 0.433i)5-s + (−0.511 − 1.08i)6-s + (−0.301 + 0.415i)7-s + (0.242 + 0.257i)8-s + (1.74 − 0.692i)9-s + (−0.455 − 0.541i)10-s + (−1.75 + 0.450i)11-s + (−0.653 + 0.540i)12-s + (0.171 + 0.432i)13-s + (0.337 + 0.133i)14-s + (1.36 − 1.00i)15-s + (0.133 − 0.211i)16-s + (−0.275 + 0.500i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.492 + 0.870i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61707 - 0.943299i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61707 - 0.943299i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.248 + 0.968i)T \) |
| 5 | \( 1 + (-2.01 + 0.968i)T \) |
| good | 3 | \( 1 + (-2.88 + 0.550i)T + (2.78 - 1.10i)T^{2} \) |
| 7 | \( 1 + (0.797 - 1.09i)T + (-2.16 - 6.65i)T^{2} \) |
| 11 | \( 1 + (5.82 - 1.49i)T + (9.63 - 5.29i)T^{2} \) |
| 13 | \( 1 + (-0.617 - 1.55i)T + (-9.47 + 8.89i)T^{2} \) |
| 17 | \( 1 + (1.13 - 2.06i)T + (-9.10 - 14.3i)T^{2} \) |
| 19 | \( 1 + (-0.142 + 0.746i)T + (-17.6 - 6.99i)T^{2} \) |
| 23 | \( 1 + (7.59 + 0.477i)T + (22.8 + 2.88i)T^{2} \) |
| 29 | \( 1 + (-3.73 - 0.471i)T + (28.0 + 7.21i)T^{2} \) |
| 31 | \( 1 + (1.31 + 0.724i)T + (16.6 + 26.1i)T^{2} \) |
| 37 | \( 1 + (5.99 + 3.80i)T + (15.7 + 33.4i)T^{2} \) |
| 41 | \( 1 + (0.446 + 7.10i)T + (-40.6 + 5.13i)T^{2} \) |
| 43 | \( 1 + (-11.3 + 3.67i)T + (34.7 - 25.2i)T^{2} \) |
| 47 | \( 1 + (-1.23 + 1.31i)T + (-2.95 - 46.9i)T^{2} \) |
| 53 | \( 1 + (2.74 + 1.29i)T + (33.7 + 40.8i)T^{2} \) |
| 59 | \( 1 + (-2.82 - 3.41i)T + (-11.0 + 57.9i)T^{2} \) |
| 61 | \( 1 + (0.860 - 13.6i)T + (-60.5 - 7.64i)T^{2} \) |
| 67 | \( 1 + (-0.0237 - 0.188i)T + (-64.8 + 16.6i)T^{2} \) |
| 71 | \( 1 + (-0.464 - 0.436i)T + (4.45 + 70.8i)T^{2} \) |
| 73 | \( 1 + (-9.25 - 7.65i)T + (13.6 + 71.7i)T^{2} \) |
| 79 | \( 1 + (-2.59 - 13.6i)T + (-73.4 + 29.0i)T^{2} \) |
| 83 | \( 1 + (6.68 + 1.27i)T + (77.1 + 30.5i)T^{2} \) |
| 89 | \( 1 + (-9.38 + 11.3i)T + (-16.6 - 87.4i)T^{2} \) |
| 97 | \( 1 + (-1.37 + 10.8i)T + (-93.9 - 24.1i)T^{2} \) |
| 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
−12.40878654554105904913241841928, −10.54018543859848288013260365923, −9.880832101332188390541983384596, −8.956199226386491558590944018723, −8.358543707537450359713833442782, −7.30998665028010817301272958951, −5.65543997649065175593632627546, −4.14001229267512148067366771889, −2.62048776863414381619348401042, −2.03086593462148838856261944652,
2.37492660933537119379858596062, 3.42836322237772678527381156933, 5.01449975130913169444029929822, 6.31342823181885027068889356312, 7.64754877913998272903387002684, 8.166336275565962316141144621234, 9.287046912288174338871199652369, 10.07086651597167844494383331343, 10.64183114258539812830667344100, 12.80371018939981534475681956761
