| L(s) = 1 | + (−1.57 + 0.714i)3-s + 1.85i·5-s + 0.707i·7-s + (1.97 − 2.25i)9-s − 4.63·11-s − 1.78·13-s + (−1.32 − 2.91i)15-s + 1.47i·17-s + i·19-s + (−0.505 − 1.11i)21-s − 2.68·23-s + 1.57·25-s + (−1.51 + 4.97i)27-s − 2.44i·29-s − 6.63i·31-s + ⋯ |
| L(s) = 1 | + (−0.910 + 0.412i)3-s + 0.827i·5-s + 0.267i·7-s + (0.659 − 0.751i)9-s − 1.39·11-s − 0.494·13-s + (−0.341 − 0.753i)15-s + 0.358i·17-s + 0.229i·19-s + (−0.110 − 0.243i)21-s − 0.559·23-s + 0.315·25-s + (−0.290 + 0.956i)27-s − 0.453i·29-s − 1.19i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.582 + 0.812i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0186913 - 0.0363956i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0186913 - 0.0363956i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.57 - 0.714i)T \) |
| 19 | \( 1 - iT \) |
| good | 5 | \( 1 - 1.85iT - 5T^{2} \) |
| 7 | \( 1 - 0.707iT - 7T^{2} \) |
| 11 | \( 1 + 4.63T + 11T^{2} \) |
| 13 | \( 1 + 1.78T + 13T^{2} \) |
| 17 | \( 1 - 1.47iT - 17T^{2} \) |
| 23 | \( 1 + 2.68T + 23T^{2} \) |
| 29 | \( 1 + 2.44iT - 29T^{2} \) |
| 31 | \( 1 + 6.63iT - 31T^{2} \) |
| 37 | \( 1 + 3.04T + 37T^{2} \) |
| 41 | \( 1 + 4.95iT - 41T^{2} \) |
| 43 | \( 1 + 6.80iT - 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 + 3.89iT - 53T^{2} \) |
| 59 | \( 1 + 8.14T + 59T^{2} \) |
| 61 | \( 1 + 11.9T + 61T^{2} \) |
| 67 | \( 1 + 4.84iT - 67T^{2} \) |
| 71 | \( 1 - 6.09T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 2.21iT - 79T^{2} \) |
| 83 | \( 1 - 2.65T + 83T^{2} \) |
| 89 | \( 1 + 10.9iT - 89T^{2} \) |
| 97 | \( 1 + 9.45T + 97T^{2} \) |
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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
−10.22045589180438700907456705355, −9.158071303055519079471455294564, −7.943434435199061942549101477531, −7.19237062815605567300220335399, −6.19887897448914731317820732334, −5.50113225996287218029024652850, −4.57464444853593615307748305118, −3.39376194091474647302125017192, −2.21343297828125477655371735620, −0.02178341831154577098022560703,
1.41438663293719526843537893690, 2.86991083533696236791450757852, 4.62043684250207362171974480161, 5.02621750703964960325088420838, 5.95972296512921233199700848956, 7.04437122250150371510256656687, 7.75099799567384206350677109149, 8.589598267497443937516600681034, 9.717086802081990869544294650540, 10.48202315796750458575569016668
