| L(s) = 1 | + 2.30i·5-s − 4.59i·7-s + 3.79·11-s − 2.59·13-s − 3.13i·17-s − i·19-s − 3.59·23-s − 0.333·25-s + 7.15i·29-s − 6.83i·31-s + 10.6·35-s − 5.26·37-s − 10.9i·41-s − 5.05i·43-s − 2.74·47-s + ⋯ |
| L(s) = 1 | + 1.03i·5-s − 1.73i·7-s + 1.14·11-s − 0.719·13-s − 0.761i·17-s − 0.229i·19-s − 0.750·23-s − 0.0666·25-s + 1.32i·29-s − 1.22i·31-s + 1.79·35-s − 0.864·37-s − 1.71i·41-s − 0.771i·43-s − 0.400·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0917 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.339201814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.339201814\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + iT \) |
| good | 5 | \( 1 - 2.30iT - 5T^{2} \) |
| 7 | \( 1 + 4.59iT - 7T^{2} \) |
| 11 | \( 1 - 3.79T + 11T^{2} \) |
| 13 | \( 1 + 2.59T + 13T^{2} \) |
| 17 | \( 1 + 3.13iT - 17T^{2} \) |
| 23 | \( 1 + 3.59T + 23T^{2} \) |
| 29 | \( 1 - 7.15iT - 29T^{2} \) |
| 31 | \( 1 + 6.83iT - 31T^{2} \) |
| 37 | \( 1 + 5.26T + 37T^{2} \) |
| 41 | \( 1 + 10.9iT - 41T^{2} \) |
| 43 | \( 1 + 5.05iT - 43T^{2} \) |
| 47 | \( 1 + 2.74T + 47T^{2} \) |
| 53 | \( 1 + 0.137iT - 53T^{2} \) |
| 59 | \( 1 - 3.11T + 59T^{2} \) |
| 61 | \( 1 + 1.56T + 61T^{2} \) |
| 67 | \( 1 - 2.53iT - 67T^{2} \) |
| 71 | \( 1 - 4.05T + 71T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 - 11.5iT - 79T^{2} \) |
| 83 | \( 1 + 7.81T + 83T^{2} \) |
| 89 | \( 1 + 6.79iT - 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.640199451681018806111548247366, −7.51035223813553681154265568095, −7.04183032388721638612433060245, −6.74624104578396693320060998911, −5.57643124925037616394263891297, −4.45720527171354847335945538405, −3.82333384303866377201281639608, −3.03412685922549071592176795818, −1.77935946578297554254532124667, −0.43692684463005116990459064482,
1.37321941388885770130778946085, 2.23687097306042712529900144894, 3.36620797074788143147749034787, 4.49363971664759225711751744775, 5.09065133592114187247274753605, 6.01194953759811138740482100344, 6.44316892167082877969449081173, 7.75015669209441782069500883806, 8.490509942803360635894432650085, 8.915695443856678411205500973410
