| L(s) = 1 | + (4.5 + 7.79i)3-s + (−3 + 5.19i)5-s + (−40.5 + 70.1i)9-s + (54 + 93.5i)11-s − 346·13-s − 54·15-s + (699 + 1.21e3i)17-s + (506 − 876. i)19-s + (768 − 1.33e3i)23-s + (1.54e3 + 2.67e3i)25-s − 729·27-s − 3.76e3·29-s + (368 + 637. i)31-s + (−486 + 841. i)33-s + (−1.02e3 + 1.77e3i)37-s + ⋯ |
| L(s) = 1 | + (0.288 + 0.499i)3-s + (−0.0536 + 0.0929i)5-s + (−0.166 + 0.288i)9-s + (0.134 + 0.233i)11-s − 0.567·13-s − 0.0619·15-s + (0.586 + 1.01i)17-s + (0.321 − 0.556i)19-s + (0.302 − 0.524i)23-s + (0.494 + 0.856i)25-s − 0.192·27-s − 0.830·29-s + (0.0687 + 0.119i)31-s + (−0.0776 + 0.134i)33-s + (−0.123 + 0.213i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.9981365620\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9981365620\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.5 - 7.79i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + (3 - 5.19i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (-54 - 93.5i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + 346T + 3.71e5T^{2} \) |
| 17 | \( 1 + (-699 - 1.21e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-506 + 876. i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-768 + 1.33e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-368 - 637. i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.02e3 - 1.77e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.55e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (2.28e3 - 3.94e3i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-3.98e3 - 6.89e3i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (-3.51e3 - 6.07e3i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.34e4 - 2.32e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (2.60e4 + 4.51e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 2.54e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + (-4.88e3 - 8.45e3i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (3.43e4 - 5.94e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + 6.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (-2.07e4 + 3.59e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 1.11e5T + 8.58e9T^{2} \) |
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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
−10.30768616017024427981387059803, −9.458380857217662344581914987080, −8.700581327075073025105368669581, −7.70714624427221395292346851693, −6.86024127506645067373861994751, −5.65413034633560917057799552898, −4.74500165619026516235106025757, −3.70386519203619722944042202616, −2.71585332982859187762490226605, −1.40417024146723616154567069019,
0.20817287757675566114708184526, 1.38031929112813711367364517213, 2.61783149160094153102135682315, 3.60532491933222595242205452106, 4.91447959557260422094010899444, 5.84330521272567582879225213185, 6.98877939840209673341032484440, 7.63733509769275300760698128395, 8.576406273930600003478011070247, 9.459658367681639021872237564821
