| L(s) = 1 | − 2.23i·5-s + 11.3i·7-s + 21.3i·11-s − 21.8·13-s + 11.7i·17-s + 24.6i·19-s + (12.9 + 19.0i)23-s − 5.00·25-s + 0.148·29-s − 49.8·31-s + 25.3·35-s + 46.1i·37-s − 4.65·41-s − 19.0i·43-s − 6.26·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 1.61i·7-s + 1.94i·11-s − 1.68·13-s + 0.693i·17-s + 1.29i·19-s + (0.562 + 0.827i)23-s − 0.200·25-s + 0.00510·29-s − 1.60·31-s + 0.723·35-s + 1.24i·37-s − 0.113·41-s − 0.442i·43-s − 0.133·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.562i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.827 + 0.562i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.063664062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.063664062\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-12.9 - 19.0i)T \) |
| good | 7 | \( 1 - 11.3iT - 49T^{2} \) |
| 11 | \( 1 - 21.3iT - 121T^{2} \) |
| 13 | \( 1 + 21.8T + 169T^{2} \) |
| 17 | \( 1 - 11.7iT - 289T^{2} \) |
| 19 | \( 1 - 24.6iT - 361T^{2} \) |
| 29 | \( 1 - 0.148T + 841T^{2} \) |
| 31 | \( 1 + 49.8T + 961T^{2} \) |
| 37 | \( 1 - 46.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 4.65T + 1.68e3T^{2} \) |
| 43 | \( 1 + 19.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 6.26T + 2.20e3T^{2} \) |
| 53 | \( 1 - 61.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 28.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 13.1iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 105. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 38.8T + 5.04e3T^{2} \) |
| 73 | \( 1 - 41.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 44.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 4.07iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 46.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 132. iT - 9.40e3T^{2} \) |
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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
−8.777818459823190005098949428086, −7.85199471232738026913464802273, −7.38511761609408068902422432751, −6.46885806692798543362460876877, −5.49365985689470370633969582780, −5.09920911225331843070718075797, −4.33143108389791247419232989680, −3.18743119971667590182475957547, −2.05004999995765198729150824301, −1.79591255288811922857815417172,
0.28347640521971878660384075270, 0.74299791916259656552555673510, 2.36364718904948429904145464820, 3.15152567194025052568347462466, 3.89726969370638965125558468587, 4.81847526227395004243844309435, 5.49323639465644154662631569460, 6.60110679903747937996357505878, 7.13878142252897465078707923863, 7.57052834522685426015975975237
