| L(s) = 1 | + (−0.776 − 0.504i)2-s + (−1.23 − 0.473i)3-s + (−0.464 − 1.04i)4-s + (2.01 − 0.972i)5-s + (0.718 + 0.989i)6-s + (−2.64 − 0.0993i)7-s + (−0.455 + 2.87i)8-s + (−0.932 − 0.839i)9-s + (−2.05 − 0.260i)10-s + (−0.526 − 0.584i)11-s + (0.0790 + 1.50i)12-s + (−0.371 + 0.729i)13-s + (2.00 + 1.41i)14-s + (−2.94 + 0.246i)15-s + (0.272 − 0.303i)16-s + (−5.77 − 4.67i)17-s + ⋯ |
| L(s) = 1 | + (−0.549 − 0.356i)2-s + (−0.712 − 0.273i)3-s + (−0.232 − 0.522i)4-s + (0.900 − 0.434i)5-s + (0.293 + 0.403i)6-s + (−0.999 − 0.0375i)7-s + (−0.160 + 1.01i)8-s + (−0.310 − 0.279i)9-s + (−0.649 − 0.0822i)10-s + (−0.158 − 0.176i)11-s + (0.0228 + 0.435i)12-s + (−0.103 + 0.202i)13-s + (0.535 + 0.376i)14-s + (−0.760 + 0.0635i)15-s + (0.0682 − 0.0757i)16-s + (−1.40 − 1.13i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.981 + 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0410353 - 0.423460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0410353 - 0.423460i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-2.01 + 0.972i)T \) |
| 7 | \( 1 + (2.64 + 0.0993i)T \) |
| good | 2 | \( 1 + (0.776 + 0.504i)T + (0.813 + 1.82i)T^{2} \) |
| 3 | \( 1 + (1.23 + 0.473i)T + (2.22 + 2.00i)T^{2} \) |
| 11 | \( 1 + (0.526 + 0.584i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.371 - 0.729i)T + (-7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (5.77 + 4.67i)T + (3.53 + 16.6i)T^{2} \) |
| 19 | \( 1 + (1.68 + 0.748i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (2.22 - 3.42i)T + (-9.35 - 21.0i)T^{2} \) |
| 29 | \( 1 + (-5.50 + 7.57i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.69 + 0.493i)T + (30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (-6.55 + 0.343i)T + (36.7 - 3.86i)T^{2} \) |
| 41 | \( 1 + (2.40 - 0.783i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-1.79 + 1.79i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.25 + 8.95i)T + (-9.77 + 45.9i)T^{2} \) |
| 53 | \( 1 + (-3.22 + 8.40i)T + (-39.3 - 35.4i)T^{2} \) |
| 59 | \( 1 + (11.1 + 2.36i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-1.75 - 8.26i)T + (-55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.574 + 0.709i)T + (-13.9 - 65.5i)T^{2} \) |
| 71 | \( 1 + (-0.767 - 0.557i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.194 + 3.70i)T + (-72.6 - 7.63i)T^{2} \) |
| 79 | \( 1 + (-4.86 - 0.511i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-5.14 - 0.814i)T + (78.9 + 25.6i)T^{2} \) |
| 89 | \( 1 + (-0.565 + 0.120i)T + (81.3 - 36.1i)T^{2} \) |
| 97 | \( 1 + (10.5 - 1.67i)T + (92.2 - 29.9i)T^{2} \) |
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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
−12.04175540301448729076158111821, −11.23551352229291893971537056827, −10.06504759916362971171791231822, −9.450979534169160221929879774709, −8.569388936407385534238447300362, −6.61295906395154233134259422831, −5.96493702676435030932599387454, −4.80422816580110185310532002201, −2.44714316366380906054947601811, −0.48633954437755653950909955555,
2.81162872773623468407336251126, 4.52135543325687860334757880653, 6.15967707337950150349589886500, 6.63929728744362182722820817075, 8.198985683944290130472572887905, 9.190313825213628018887902865103, 10.21153482248867873614723663925, 10.83624129716609672050972386346, 12.38450910201072111819595351457, 13.02072095183989300424323549703
