| L(s) = 1 | − 2-s + 4-s + (−1.62 + 1.53i)5-s + 1.22i·7-s − 8-s + (1.62 − 1.53i)10-s − 1.87·11-s − 6.50·13-s − 1.22i·14-s + 16-s − 0.765·17-s + 3.34i·19-s + (−1.62 + 1.53i)20-s + 1.87·22-s + 1.38·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.728 + 0.685i)5-s + 0.461i·7-s − 0.353·8-s + (0.515 − 0.484i)10-s − 0.564·11-s − 1.80·13-s − 0.326i·14-s + 0.250·16-s − 0.185·17-s + 0.767i·19-s + (−0.364 + 0.342i)20-s + 0.398·22-s + 0.289·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3112562283\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3112562283\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.62 - 1.53i)T \) |
| 37 | \( 1 + (1.76 - 5.82i)T \) |
| good | 7 | \( 1 - 1.22iT - 7T^{2} \) |
| 11 | \( 1 + 1.87T + 11T^{2} \) |
| 13 | \( 1 + 6.50T + 13T^{2} \) |
| 17 | \( 1 + 0.765T + 17T^{2} \) |
| 19 | \( 1 - 3.34iT - 19T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 - 1.72iT - 29T^{2} \) |
| 31 | \( 1 - 4.11iT - 31T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 + 4.91T + 43T^{2} \) |
| 47 | \( 1 + 6.30iT - 47T^{2} \) |
| 53 | \( 1 + 2.57iT - 53T^{2} \) |
| 59 | \( 1 + 10.5iT - 59T^{2} \) |
| 61 | \( 1 + 11.1iT - 61T^{2} \) |
| 67 | \( 1 - 11.1iT - 67T^{2} \) |
| 71 | \( 1 + 0.963T + 71T^{2} \) |
| 73 | \( 1 + 9.03iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 0.00656iT - 83T^{2} \) |
| 89 | \( 1 - 4.70iT - 89T^{2} \) |
| 97 | \( 1 - 0.403T + 97T^{2} \) |
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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
−8.312425344475881063563503045979, −7.88570700776468237338411869412, −7.03318132971573570017930816792, −6.64276239487563022633201267053, −5.41859331294178023607703190632, −4.77582284704709547737731407747, −3.51904840836068745621573205813, −2.76433416821006564663851462899, −1.92698223429742872847518228803, −0.17094513116071586305678012135,
0.73135391295153766082590316641, 2.14187417609577263321463063852, 3.03008928669094129494085945541, 4.23832035659824473733750856739, 4.85783136332153685030285783521, 5.68499696712403721644938640983, 6.91371742361182270746738086800, 7.46489340990739059171462385885, 7.87173274533076305896873122475, 8.856769661605637367115321305134
