| L(s) = 1 | + 2.23·5-s − 9.06i·7-s − 4.28i·11-s + 9.41·13-s − 18·17-s − 36.2i·19-s − 22.9i·23-s + 5.00·25-s − 44.8·29-s + 35.2i·31-s − 20.2i·35-s − 6.58·37-s − 52.2·41-s − 28.8i·43-s + 90.1i·47-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.29i·7-s − 0.389i·11-s + 0.724·13-s − 1.05·17-s − 1.90i·19-s − 0.996i·23-s + 0.200·25-s − 1.54·29-s + 1.13i·31-s − 0.579i·35-s − 0.177·37-s − 1.27·41-s − 0.670i·43-s + 1.91i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9757995000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9757995000\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 + 9.06iT - 49T^{2} \) |
| 11 | \( 1 + 4.28iT - 121T^{2} \) |
| 13 | \( 1 - 9.41T + 169T^{2} \) |
| 17 | \( 1 + 18T + 289T^{2} \) |
| 19 | \( 1 + 36.2iT - 361T^{2} \) |
| 23 | \( 1 + 22.9iT - 529T^{2} \) |
| 29 | \( 1 + 44.8T + 841T^{2} \) |
| 31 | \( 1 - 35.2iT - 961T^{2} \) |
| 37 | \( 1 + 6.58T + 1.36e3T^{2} \) |
| 41 | \( 1 + 52.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + 28.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 90.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 52.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 17.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 50.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 33.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 20.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 91.6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 42.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 22.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 47.6T + 7.92e3T^{2} \) |
| 97 | \( 1 + 160.T + 9.40e3T^{2} \) |
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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
−8.410280602088192052986720999514, −7.14156803877297564328628264724, −6.91586272308711974274544226918, −6.02968192120810124911067705468, −4.99514720986612031215483526282, −4.30424039541895156739127586908, −3.43043369770199050635635711910, −2.40337319295983129064783712190, −1.21463924513760570638580796774, −0.21472915738000314438507381759,
1.66909179841221306004049396786, 2.15811365234080246760338749448, 3.39935649499350513456517797454, 4.18692551626211486161611683133, 5.57332254389958362298680547885, 5.62399452565668324170255065268, 6.58453914873536851234397847439, 7.50132356475155383359438066467, 8.414692614717888268338056965671, 8.884360511206473770134535192504
