| L(s) = 1 | + (3.87 + 3.46i)3-s + 8.94·5-s − 7.74i·7-s + (3.00 + 26.8i)9-s + 34.6i·11-s − 10i·13-s + (34.6 + 30.9i)15-s + 35.7i·17-s + 69.7·19-s + (26.8 − 30.0i)21-s − 96.9·23-s − 44.9·25-s + (−81.3 + 114. i)27-s + 152.·29-s + 224. i·31-s + ⋯ |
| L(s) = 1 | + (0.745 + 0.666i)3-s + 0.799·5-s − 0.418i·7-s + (0.111 + 0.993i)9-s + 0.949i·11-s − 0.213i·13-s + (0.596 + 0.533i)15-s + 0.510i·17-s + 0.841·19-s + (0.278 − 0.311i)21-s − 0.879·23-s − 0.359·25-s + (−0.579 + 0.814i)27-s + 0.973·29-s + 1.30i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0556 - 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0556 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.944714479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.944714479\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-3.87 - 3.46i)T \) |
| good | 5 | \( 1 - 8.94T + 125T^{2} \) |
| 7 | \( 1 + 7.74iT - 343T^{2} \) |
| 11 | \( 1 - 34.6iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 10iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 35.7iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 69.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 152.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 224. iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 130iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 125. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 193.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 545.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 173. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 442iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 735.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 410T + 3.89e5T^{2} \) |
| 79 | \( 1 + 85.2iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 840. iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 770T + 9.12e5T^{2} \) |
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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
−10.13789774556265608653420995093, −9.470001207773290080814313171075, −8.526837476854205188889422394349, −7.69236822617952786521567330738, −6.74116513425336993458544233814, −5.54973754397508633122840939892, −4.66505599035775836346527785678, −3.68869052419302128247290635168, −2.54405152122883090771710911091, −1.48804247753461022025947677208,
0.71085950296386130832905792695, 2.00787118761106088620349553122, 2.84311664757298957907149873368, 3.98499029626208439325922812298, 5.54863028253765323341382894059, 6.12287030607002163339544954083, 7.16634931930807539440871260354, 8.055399217673525421226197030573, 8.880225720182655089451124420749, 9.522809680014970238491522997780
