| L(s) = 1 | − 5·2-s − 7·4-s − 94·5-s − 49·7-s + 195·8-s + 470·10-s − 52·11-s − 770·13-s + 245·14-s − 751·16-s + 2.02e3·17-s + 1.73e3·19-s + 658·20-s + 260·22-s + 576·23-s + 5.71e3·25-s + 3.85e3·26-s + 343·28-s − 5.51e3·29-s + 6.33e3·31-s − 2.48e3·32-s − 1.01e4·34-s + 4.60e3·35-s − 7.33e3·37-s − 8.66e3·38-s − 1.83e4·40-s + 3.26e3·41-s + ⋯ |
| L(s) = 1 | − 0.883·2-s − 0.218·4-s − 1.68·5-s − 0.377·7-s + 1.07·8-s + 1.48·10-s − 0.129·11-s − 1.26·13-s + 0.334·14-s − 0.733·16-s + 1.69·17-s + 1.10·19-s + 0.367·20-s + 0.114·22-s + 0.227·23-s + 1.82·25-s + 1.11·26-s + 0.0826·28-s − 1.21·29-s + 1.18·31-s − 0.428·32-s − 1.49·34-s + 0.635·35-s − 0.881·37-s − 0.972·38-s − 1.81·40-s + 0.303·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4952802083\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4952802083\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 2 | \( 1 + 5 T + p^{5} T^{2} \) |
| 5 | \( 1 + 94 T + p^{5} T^{2} \) |
| 11 | \( 1 + 52 T + p^{5} T^{2} \) |
| 13 | \( 1 + 770 T + p^{5} T^{2} \) |
| 17 | \( 1 - 2022 T + p^{5} T^{2} \) |
| 19 | \( 1 - 1732 T + p^{5} T^{2} \) |
| 23 | \( 1 - 576 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5518 T + p^{5} T^{2} \) |
| 31 | \( 1 - 6336 T + p^{5} T^{2} \) |
| 37 | \( 1 + 7338 T + p^{5} T^{2} \) |
| 41 | \( 1 - 3262 T + p^{5} T^{2} \) |
| 43 | \( 1 - 5420 T + p^{5} T^{2} \) |
| 47 | \( 1 + 864 T + p^{5} T^{2} \) |
| 53 | \( 1 + 4182 T + p^{5} T^{2} \) |
| 59 | \( 1 - 11220 T + p^{5} T^{2} \) |
| 61 | \( 1 + 45602 T + p^{5} T^{2} \) |
| 67 | \( 1 - 1396 T + p^{5} T^{2} \) |
| 71 | \( 1 + 18720 T + p^{5} T^{2} \) |
| 73 | \( 1 - 46362 T + p^{5} T^{2} \) |
| 79 | \( 1 - 97424 T + p^{5} T^{2} \) |
| 83 | \( 1 - 81228 T + p^{5} T^{2} \) |
| 89 | \( 1 - 3182 T + p^{5} T^{2} \) |
| 97 | \( 1 - 4914 T + p^{5} T^{2} \) |
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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
−14.11219261848242256852287043776, −12.54059745056699643476025185018, −11.70674584951593231963617299380, −10.32476954127726507066844377849, −9.290822125624696675519937823595, −7.85609523825536629722700599680, −7.42012405406319714663150973781, −4.95114625674260926301665454186, −3.46024921352335639525952928859, −0.63869992670379025229142949801,
0.63869992670379025229142949801, 3.46024921352335639525952928859, 4.95114625674260926301665454186, 7.42012405406319714663150973781, 7.85609523825536629722700599680, 9.290822125624696675519937823595, 10.32476954127726507066844377849, 11.70674584951593231963617299380, 12.54059745056699643476025185018, 14.11219261848242256852287043776
