| L(s) = 1 | + i·3-s + 4i·7-s − 9-s − 2i·13-s + 6i·17-s − 4·19-s − 4·21-s − i·27-s + 6·29-s − 8·31-s + 2i·37-s + 2·39-s − 6·41-s − 4i·43-s − 9·49-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.51i·7-s − 0.333·9-s − 0.554i·13-s + 1.45i·17-s − 0.917·19-s − 0.872·21-s − 0.192i·27-s + 1.11·29-s − 1.43·31-s + 0.328i·37-s + 0.320·39-s − 0.937·41-s − 0.609i·43-s − 1.28·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.059979109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.059979109\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 2iT - 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + 18T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−10.19174573345121234503406816936, −9.102583885333548575319203067640, −8.634492796115289623641541591247, −7.923960150013263745472702459032, −6.52816372776241258618775144025, −5.82500079550649568890455986869, −5.09997914259797008869091954507, −4.00079934849080517680346014977, −2.95446901053280278700842005809, −1.90070444860112036439956286397,
0.43816537375513400236523009121, 1.75916814902390267633954093228, 3.11151715435156831813944606012, 4.19621709129075032138817413281, 5.00958422924308679007974735678, 6.31622233499071781882115027900, 7.03106501189624728289311285097, 7.53290580843691848524365423607, 8.513959693519003072099003833275, 9.412415368895010110577584049983
