L(s) = 1 | + (1.5 − 2.59i)3-s + (−2 − 3.46i)5-s + (−4.5 − 7.79i)9-s + (10 − 17.3i)11-s − 4·13-s − 12·15-s + (−12 + 20.7i)17-s + (−22 − 38.1i)19-s + (−36 − 62.3i)23-s + (54.5 − 94.3i)25-s − 27·27-s − 38·29-s + (−92 + 159. i)31-s + (−30.0 − 51.9i)33-s + (15 + 25.9i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.178 − 0.309i)5-s + (−0.166 − 0.288i)9-s + (0.274 − 0.474i)11-s − 0.0853·13-s − 0.206·15-s + (−0.171 + 0.296i)17-s + (−0.265 − 0.460i)19-s + (−0.326 − 0.565i)23-s + (0.435 − 0.755i)25-s − 0.192·27-s − 0.243·29-s + (−0.533 + 0.923i)31-s + (−0.158 − 0.274i)33-s + (0.0666 + 0.115i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9088050282\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9088050282\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 2.59i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (2 + 3.46i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (-10 + 17.3i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (12 - 20.7i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (22 + 38.1i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (36 + 62.3i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 38T + 2.43e4T^{2} \) |
| 31 | \( 1 + (92 - 159. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-15 - 25.9i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 + 216T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (260 + 450. i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-73 + 126. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (230 - 398. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (314 + 543. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (278 - 481. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 592T + 3.57e5T^{2} \) |
| 73 | \( 1 + (512 - 886. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-52 - 90.0i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 324T + 5.71e5T^{2} \) |
| 89 | \( 1 + (448 + 775. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 920T + 9.12e5T^{2} \) |
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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
−9.834085229446507368539555345567, −8.658687213917785721874352635237, −8.369918648103572214385891976285, −7.10973439886724360257590309703, −6.39978513444081223578266946413, −5.23184365901608696085862892931, −4.09685392771783319742378103557, −2.93317078867732140898268362989, −1.63085911624349848047001637523, −0.24657270150935342364167361983,
1.73220998272075380163213063418, 3.07410585793447393389747772279, 4.04498022636050785300350339547, 5.05388729976335712308901624423, 6.17938443561965556447615933278, 7.26128547048016974760607559497, 8.028641285827839899843377539766, 9.152337281683872213218539948756, 9.730072777810793752716103066760, 10.69796463689668568524506496322