| L(s) = 1 | + (31.1 − 5.48i)3-s + (−167. + 140. i)5-s + (273. − 473. i)7-s + (253. − 92.2i)9-s + (1.16e3 + 2.01e3i)11-s + (2.46e3 + 434. i)13-s + (−4.43e3 + 5.27e3i)15-s + (3.13e3 + 1.14e3i)17-s + (−32.2 + 6.85e3i)19-s + (5.91e3 − 1.62e4i)21-s + (1.19e3 + 9.99e2i)23-s + (5.54e3 − 3.14e4i)25-s + (−1.25e4 + 7.25e3i)27-s + (1.28e4 + 3.54e4i)29-s + (3.59e3 + 2.07e3i)31-s + ⋯ |
| L(s) = 1 | + (1.15 − 0.203i)3-s + (−1.33 + 1.12i)5-s + (0.797 − 1.38i)7-s + (0.347 − 0.126i)9-s + (0.873 + 1.51i)11-s + (1.12 + 0.197i)13-s + (−1.31 + 1.56i)15-s + (0.638 + 0.232i)17-s + (−0.00470 + 0.999i)19-s + (0.638 − 1.75i)21-s + (0.0979 + 0.0821i)23-s + (0.354 − 2.01i)25-s + (−0.638 + 0.368i)27-s + (0.528 + 1.45i)29-s + (0.120 + 0.0696i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.733 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.28744 + 0.897167i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.28744 + 0.897167i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (32.2 - 6.85e3i)T \) |
| good | 3 | \( 1 + (-31.1 + 5.48i)T + (685. - 249. i)T^{2} \) |
| 5 | \( 1 + (167. - 140. i)T + (2.71e3 - 1.53e4i)T^{2} \) |
| 7 | \( 1 + (-273. + 473. i)T + (-5.88e4 - 1.01e5i)T^{2} \) |
| 11 | \( 1 + (-1.16e3 - 2.01e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-2.46e3 - 434. i)T + (4.53e6 + 1.65e6i)T^{2} \) |
| 17 | \( 1 + (-3.13e3 - 1.14e3i)T + (1.84e7 + 1.55e7i)T^{2} \) |
| 23 | \( 1 + (-1.19e3 - 9.99e2i)T + (2.57e7 + 1.45e8i)T^{2} \) |
| 29 | \( 1 + (-1.28e4 - 3.54e4i)T + (-4.55e8 + 3.82e8i)T^{2} \) |
| 31 | \( 1 + (-3.59e3 - 2.07e3i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + 3.07e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (4.85e4 - 8.56e3i)T + (4.46e9 - 1.62e9i)T^{2} \) |
| 43 | \( 1 + (-5.09e4 + 4.27e4i)T + (1.09e9 - 6.22e9i)T^{2} \) |
| 47 | \( 1 + (-4.54e4 + 1.65e4i)T + (8.25e9 - 6.92e9i)T^{2} \) |
| 53 | \( 1 + (-8.10e4 + 9.66e4i)T + (-3.84e9 - 2.18e10i)T^{2} \) |
| 59 | \( 1 + (-3.24e4 + 8.90e4i)T + (-3.23e10 - 2.71e10i)T^{2} \) |
| 61 | \( 1 + (-1.62e4 - 1.36e4i)T + (8.94e9 + 5.07e10i)T^{2} \) |
| 67 | \( 1 + (-1.00e5 - 2.75e5i)T + (-6.92e10 + 5.81e10i)T^{2} \) |
| 71 | \( 1 + (3.34e5 + 3.98e5i)T + (-2.22e10 + 1.26e11i)T^{2} \) |
| 73 | \( 1 + (1.04e5 + 5.92e5i)T + (-1.42e11 + 5.17e10i)T^{2} \) |
| 79 | \( 1 + (4.00e5 - 7.05e4i)T + (2.28e11 - 8.31e10i)T^{2} \) |
| 83 | \( 1 + (3.86e5 - 6.69e5i)T + (-1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + (1.10e5 + 1.94e4i)T + (4.67e11 + 1.69e11i)T^{2} \) |
| 97 | \( 1 + (2.97e5 - 8.17e5i)T + (-6.38e11 - 5.35e11i)T^{2} \) |
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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
−13.88093959093090593757897572656, −12.27412435283638573311206109887, −11.14160643623903021146030249616, −10.21507223949949323907992432686, −8.513397253876566858356468054940, −7.56165441557015465875971182834, −6.97087575460798083899690903449, −4.13982598911497564989580892227, −3.48529147205306349130150299520, −1.53592539720518263213164775567,
0.951765119146956055153677669712, 3.00852316495239882963329909190, 4.21225209795574709154472639243, 5.74154449767045713924532158114, 8.044834230274048414769020404472, 8.589767224775499824870400507637, 9.059394453562457525592191456310, 11.44532442136160805779411140649, 11.80995774438847897648331188996, 13.27983488470824023861013510035
