L(s) = 1 | + (24.4 + 11.3i)3-s + (124. − 8.26i)5-s − 577. i·7-s + (471. + 556. i)9-s − 1.96e3i·11-s + 2.09e3i·13-s + (3.14e3 + 1.21e3i)15-s − 1.15e3·17-s + 5.61e3·19-s + (6.55e3 − 1.41e4i)21-s − 3.38e3·23-s + (1.54e4 − 2.06e3i)25-s + (5.22e3 + 1.89e4i)27-s + 5.72e3i·29-s + 3.69e4·31-s + ⋯ |
L(s) = 1 | + (0.907 + 0.420i)3-s + (0.997 − 0.0661i)5-s − 1.68i·7-s + (0.646 + 0.763i)9-s − 1.47i·11-s + 0.951i·13-s + (0.933 + 0.359i)15-s − 0.234·17-s + 0.818·19-s + (0.707 − 1.52i)21-s − 0.278·23-s + (0.991 − 0.131i)25-s + (0.265 + 0.964i)27-s + 0.234i·29-s + 1.23·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.933 + 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.84162 - 0.528673i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.84162 - 0.528673i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-24.4 - 11.3i)T \) |
| 5 | \( 1 + (-124. + 8.26i)T \) |
good | 7 | \( 1 + 577. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 1.96e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 2.09e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + 1.15e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 5.61e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 3.38e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 5.72e3iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 3.69e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 3.26e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 5.65e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 1.13e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.57e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.09e5T + 2.21e10T^{2} \) |
| 59 | \( 1 - 2.27e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.71e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 2.19e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 5.22e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.78e3iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 7.94e5T + 2.43e11T^{2} \) |
| 83 | \( 1 + 2.13e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 1.24e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 4.92e5iT - 8.32e11T^{2} \) |
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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
−13.79571057463639109766750749600, −13.31139706119921619140994590565, −11.16033276718204935866485031611, −10.16624337768040715937162885783, −9.210015011880827766007610228792, −7.892904836721590487985322413908, −6.46987788010690202239591784004, −4.57969911277800491147822501209, −3.18359397416401529152971999468, −1.27299152717091183565495809030,
1.83709847140846487379757913636, 2.81261953029871668027981481553, 5.18519560885191925249818553533, 6.55453476951696308772850073885, 8.079647727161948952030198243768, 9.264852120164535301719938193296, 10.00105712733816044087204493164, 12.08438796038916510365948571799, 12.78194399599506225290648278017, 13.91411805381117853640041413470