| L(s) = 1 | − 3-s + 2·4-s + i·5-s + 3i·7-s + 9-s − 3i·11-s − 2·12-s + (2 + 3i)13-s − i·15-s + 4·16-s + 3·17-s + 2i·20-s − 3i·21-s − 3·23-s − 25-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 4-s + 0.447i·5-s + 1.13i·7-s + 0.333·9-s − 0.904i·11-s − 0.577·12-s + (0.554 + 0.832i)13-s − 0.258i·15-s + 16-s + 0.727·17-s + 0.447i·20-s − 0.654i·21-s − 0.625·23-s − 0.200·25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.18594 + 0.359075i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18594 + 0.359075i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-2 - 3i)T \) |
| good | 2 | \( 1 - 2T^{2} \) |
| 7 | \( 1 - 3iT - 7T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6iT - 31T^{2} \) |
| 37 | \( 1 + 9iT - 37T^{2} \) |
| 41 | \( 1 - 3iT - 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + 12iT - 47T^{2} \) |
| 53 | \( 1 + 3T + 53T^{2} \) |
| 59 | \( 1 - 12iT - 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 9iT - 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 - 15iT - 89T^{2} \) |
| 97 | \( 1 + 9iT - 97T^{2} \) |
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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
−12.21436570506772567483932616816, −11.57579243969551510243171369098, −10.96465000237073123740302955925, −9.801198713235635872882216045730, −8.539212713240407816898433411721, −7.32047176054239947683137845808, −6.14990088141191912492540793688, −5.63478094415584843436488767748, −3.56468853626112595924610399260, −2.06560962990429040857708388071,
1.42233575810623611937782673455, 3.52298040307104980526311157304, 4.99034384730193030344973823169, 6.22616628884189625318700758949, 7.25469960700158703809504228012, 8.058931741203022981873284223046, 9.866214370838282562462662110838, 10.48429516634604597866537922057, 11.41687108701045236968712565529, 12.36633564848157961815425532590
