L(s) = 1 | + (0.238 − 1.25i)3-s + (0.876 + 0.481i)4-s + (−0.809 + 0.587i)5-s + (−0.581 − 0.230i)9-s + (0.968 + 0.248i)11-s + (0.812 − 0.982i)12-s + (0.542 + 1.15i)15-s + (0.535 + 0.844i)16-s + (−0.992 + 0.125i)20-s + (−0.0235 − 0.374i)23-s + (0.309 − 0.951i)25-s + (0.255 − 0.403i)27-s + (0.110 − 0.0604i)31-s + (0.542 − 1.15i)33-s + (−0.398 − 0.481i)36-s + (0.331 + 0.521i)37-s + ⋯ |
L(s) = 1 | + (0.238 − 1.25i)3-s + (0.876 + 0.481i)4-s + (−0.809 + 0.587i)5-s + (−0.581 − 0.230i)9-s + (0.968 + 0.248i)11-s + (0.812 − 0.982i)12-s + (0.542 + 1.15i)15-s + (0.535 + 0.844i)16-s + (−0.992 + 0.125i)20-s + (−0.0235 − 0.374i)23-s + (0.309 − 0.951i)25-s + (0.255 − 0.403i)27-s + (0.110 − 0.0604i)31-s + (0.542 − 1.15i)33-s + (−0.398 − 0.481i)36-s + (0.331 + 0.521i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.910 + 0.414i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.332771780\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.332771780\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (0.809 - 0.587i)T \) |
| 11 | \( 1 + (-0.968 - 0.248i)T \) |
good | 2 | \( 1 + (-0.876 - 0.481i)T^{2} \) |
| 3 | \( 1 + (-0.238 + 1.25i)T + (-0.929 - 0.368i)T^{2} \) |
| 7 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.728 - 0.684i)T^{2} \) |
| 17 | \( 1 + (-0.535 + 0.844i)T^{2} \) |
| 19 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 23 | \( 1 + (0.0235 + 0.374i)T + (-0.992 + 0.125i)T^{2} \) |
| 29 | \( 1 + (-0.968 + 0.248i)T^{2} \) |
| 31 | \( 1 + (-0.110 + 0.0604i)T + (0.535 - 0.844i)T^{2} \) |
| 37 | \( 1 + (-0.331 - 0.521i)T + (-0.425 + 0.904i)T^{2} \) |
| 41 | \( 1 + (0.992 + 0.125i)T^{2} \) |
| 43 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 47 | \( 1 + (1.17 - 1.10i)T + (0.0627 - 0.998i)T^{2} \) |
| 53 | \( 1 + (0.824 + 1.75i)T + (-0.637 + 0.770i)T^{2} \) |
| 59 | \( 1 + (0.683 - 0.825i)T + (-0.187 - 0.982i)T^{2} \) |
| 61 | \( 1 + (0.992 - 0.125i)T^{2} \) |
| 67 | \( 1 + (-0.844 - 0.106i)T + (0.968 + 0.248i)T^{2} \) |
| 71 | \( 1 + (0.929 - 0.872i)T + (0.0627 - 0.998i)T^{2} \) |
| 73 | \( 1 + (0.187 - 0.982i)T^{2} \) |
| 79 | \( 1 + (0.929 + 0.368i)T^{2} \) |
| 83 | \( 1 + (0.929 - 0.368i)T^{2} \) |
| 89 | \( 1 + (0.929 + 1.12i)T + (-0.187 + 0.982i)T^{2} \) |
| 97 | \( 1 + (1.06 - 0.134i)T + (0.968 - 0.248i)T^{2} \) |
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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
−9.748432689929586997354862648844, −8.433160695048409761072489825444, −8.025086453629120082701990139070, −7.12354375794351490527909908395, −6.79245549841795452945208159654, −6.09125212341443570621626881985, −4.41667185859445994565288023908, −3.40287027581572162325110831083, −2.51842973315462433766319795835, −1.46247967798656412897395712023,
1.40430755150021376517611223190, 3.02993809779307383045224208495, 3.85851779901414736673386311734, 4.64923011680614854826472141569, 5.53983121195141813009192523710, 6.53421145916966520340476378928, 7.44746954537534184928960137605, 8.353825103919285186515091564330, 9.272946884573233119940358705964, 9.674672488509662250202506630982