| L(s) = 1 | − i·3-s − 4.17·5-s − 9-s − 1.71·11-s − 1.54·13-s + 4.17i·15-s + 2.33i·17-s + 7.04i·19-s − 0.468i·23-s + 12.4·25-s + i·27-s + 3.33i·29-s + 3.16·31-s + 1.71i·33-s − 8.94i·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.86·5-s − 0.333·9-s − 0.515·11-s − 0.427·13-s + 1.07i·15-s + 0.565i·17-s + 1.61i·19-s − 0.0977i·23-s + 2.48·25-s + 0.192i·27-s + 0.620i·29-s + 0.568·31-s + 0.297i·33-s − 1.47i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4934299855\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4934299855\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 4.17T + 5T^{2} \) |
| 11 | \( 1 + 1.71T + 11T^{2} \) |
| 13 | \( 1 + 1.54T + 13T^{2} \) |
| 17 | \( 1 - 2.33iT - 17T^{2} \) |
| 19 | \( 1 - 7.04iT - 19T^{2} \) |
| 23 | \( 1 + 0.468iT - 23T^{2} \) |
| 29 | \( 1 - 3.33iT - 29T^{2} \) |
| 31 | \( 1 - 3.16T + 31T^{2} \) |
| 37 | \( 1 + 8.94iT - 37T^{2} \) |
| 41 | \( 1 - 5.31iT - 41T^{2} \) |
| 43 | \( 1 - 3.42T + 43T^{2} \) |
| 47 | \( 1 + 5.90T + 47T^{2} \) |
| 53 | \( 1 + 1.56iT - 53T^{2} \) |
| 59 | \( 1 - 6.08iT - 59T^{2} \) |
| 61 | \( 1 + 9.11T + 61T^{2} \) |
| 67 | \( 1 - 7.47T + 67T^{2} \) |
| 71 | \( 1 - 3.49iT - 71T^{2} \) |
| 73 | \( 1 + 14.5iT - 73T^{2} \) |
| 79 | \( 1 + 1.68iT - 79T^{2} \) |
| 83 | \( 1 - 2.72iT - 83T^{2} \) |
| 89 | \( 1 + 2.12iT - 89T^{2} \) |
| 97 | \( 1 - 1.95iT - 97T^{2} \) |
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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
−7.946517345466098866526121083713, −7.58832437752725475412134917155, −6.83289700073698730067316016405, −5.98895247064640159669470664512, −5.06443022430057442579453178638, −4.20722621652624600530193316562, −3.56832843913403693588109377145, −2.74034823275350665821849586565, −1.47543712640999653845249150983, −0.21759296146170139115183550387,
0.69923068361326277392737473081, 2.61122422338237995101246780942, 3.18673817223645441090352422241, 4.12071939119390187454049321479, 4.71221180708376448091903557632, 5.22894955131300832591777279951, 6.56186313749281969733897491935, 7.16835065518078550086800996744, 7.87837186916937753700483700270, 8.378943357931497707464633582274
