L(s) = 1 | + (−1.63 + 1.63i)3-s + (−2.17 + 2.17i)7-s − 2.36i·9-s − 1.70·11-s + (4.43 − 4.43i)13-s + (1.63 − 1.63i)17-s + (−2.80 + 3.34i)19-s − 7.11i·21-s + (−3.24 − 3.24i)23-s + (−1.03 − 1.03i)27-s − 3.27·29-s − 7.11i·31-s + (2.80 − 2.80i)33-s + (−0.128 − 0.128i)37-s + 14.5i·39-s + ⋯ |
L(s) = 1 | + (−0.945 + 0.945i)3-s + (−0.820 + 0.820i)7-s − 0.789i·9-s − 0.515·11-s + (1.23 − 1.23i)13-s + (0.395 − 0.395i)17-s + (−0.642 + 0.766i)19-s − 1.55i·21-s + (−0.677 − 0.677i)23-s + (−0.198 − 0.198i)27-s − 0.608·29-s − 1.27i·31-s + (0.487 − 0.487i)33-s + (−0.0211 − 0.0211i)37-s + 2.32i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.143i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7736717975\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7736717975\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (2.80 - 3.34i)T \) |
good | 3 | \( 1 + (1.63 - 1.63i)T - 3iT^{2} \) |
| 7 | \( 1 + (2.17 - 2.17i)T - 7iT^{2} \) |
| 11 | \( 1 + 1.70T + 11T^{2} \) |
| 13 | \( 1 + (-4.43 + 4.43i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.63 + 1.63i)T - 17iT^{2} \) |
| 23 | \( 1 + (3.24 + 3.24i)T + 23iT^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 31 | \( 1 + 7.11iT - 31T^{2} \) |
| 37 | \( 1 + (0.128 + 0.128i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.83iT - 41T^{2} \) |
| 43 | \( 1 + (-5.24 - 5.24i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.908 - 0.908i)T - 47iT^{2} \) |
| 53 | \( 1 + (-5.47 + 5.47i)T - 53iT^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 2.63T + 61T^{2} \) |
| 67 | \( 1 + (7.71 + 7.71i)T + 67iT^{2} \) |
| 71 | \( 1 - 1.51iT - 71T^{2} \) |
| 73 | \( 1 + (-8.70 - 8.70i)T + 73iT^{2} \) |
| 79 | \( 1 + 4.78T + 79T^{2} \) |
| 83 | \( 1 + (6.46 + 6.46i)T + 83iT^{2} \) |
| 89 | \( 1 - 5.34T + 89T^{2} \) |
| 97 | \( 1 + (-10.2 - 10.2i)T + 97iT^{2} \) |
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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
−9.406777026289916459000750950731, −8.458050564950210117296407933427, −7.78456272571285914118190213997, −6.42658999550675382326595398405, −5.73664382736347267755385302780, −5.49648051027154197761085157712, −4.24419298772353970872275704561, −3.48094083205611455645429589404, −2.37739957012908621033448102263, −0.42752399962368283076900292641,
0.885876502953331651791581248510, 1.94349557876035839662169115803, 3.46144151216081887782555368226, 4.23995053845483237227617035310, 5.49982987964532989430236811293, 6.18058118843426564182490464586, 6.86587033953344461686443081941, 7.29484971377514103597923972183, 8.393787607305954324378052430275, 9.197026378929255994077326398202