| L(s) = 1 | − 8.78i·5-s − 4.31·7-s + 19.6i·11-s − 7.57·13-s + 3.65i·17-s + 33.6·19-s − 3.07i·23-s − 52.2·25-s − 24.3i·29-s + 28.8·31-s + 37.9i·35-s + 29.8·37-s − 23.0i·41-s − 8.94·43-s + 10.5i·47-s + ⋯ |
| L(s) = 1 | − 1.75i·5-s − 0.616·7-s + 1.79i·11-s − 0.582·13-s + 0.214i·17-s + 1.77·19-s − 0.133i·23-s − 2.08·25-s − 0.838i·29-s + 0.931·31-s + 1.08i·35-s + 0.805·37-s − 0.561i·41-s − 0.208·43-s + 0.224i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2916 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.637574140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637574140\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 8.78iT - 25T^{2} \) |
| 7 | \( 1 + 4.31T + 49T^{2} \) |
| 11 | \( 1 - 19.6iT - 121T^{2} \) |
| 13 | \( 1 + 7.57T + 169T^{2} \) |
| 17 | \( 1 - 3.65iT - 289T^{2} \) |
| 19 | \( 1 - 33.6T + 361T^{2} \) |
| 23 | \( 1 + 3.07iT - 529T^{2} \) |
| 29 | \( 1 + 24.3iT - 841T^{2} \) |
| 31 | \( 1 - 28.8T + 961T^{2} \) |
| 37 | \( 1 - 29.8T + 1.36e3T^{2} \) |
| 41 | \( 1 + 23.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 8.94T + 1.84e3T^{2} \) |
| 47 | \( 1 - 10.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 70.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 68.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 106.T + 3.72e3T^{2} \) |
| 67 | \( 1 + 24.6T + 4.48e3T^{2} \) |
| 71 | \( 1 + 34.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 41.8T + 5.32e3T^{2} \) |
| 79 | \( 1 - 50.5T + 6.24e3T^{2} \) |
| 83 | \( 1 - 31.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 46.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 110.T + 9.40e3T^{2} \) |
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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
−8.401483812724425618220333466244, −7.68122302250636918106406237786, −7.03830332194769843690760087708, −5.97448079496784980677058801305, −5.07973636820372605345357096449, −4.66738173986166685723285035784, −3.81907262375057021823444604570, −2.50087206502648986081198066275, −1.49407765447849931377928338925, −0.48059201376448641996071655960,
0.871966494481288806334835467515, 2.54144431466986939173464580603, 3.18737876282324882619266946537, 3.55419909674774309813971657211, 5.04283465025915526138926156942, 5.98207234209128183667264410218, 6.43289731838751138501455380759, 7.26662226544264432519024231201, 7.80187735912584277636910068950, 8.791441561423855914335703119385
