| L(s) = 1 | + 2-s − 4-s + (1 + 2i)5-s − 2i·7-s − 3·8-s + (1 + 2i)10-s − 2i·11-s − 4i·13-s − 2i·14-s − 16-s + 6·17-s + 2i·19-s + (−1 − 2i)20-s − 2i·22-s − 6i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.5·4-s + (0.447 + 0.894i)5-s − 0.755i·7-s − 1.06·8-s + (0.316 + 0.632i)10-s − 0.603i·11-s − 1.10i·13-s − 0.534i·14-s − 0.250·16-s + 1.45·17-s + 0.458i·19-s + (−0.223 − 0.447i)20-s − 0.426i·22-s − 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1305 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.009973408\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.009973408\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-1 - 2i)T \) |
| 29 | \( 1 + (-5 + 2i)T \) |
| good | 2 | \( 1 - T + 2T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 12iT - 61T^{2} \) |
| 67 | \( 1 + 6iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + 2iT - 83T^{2} \) |
| 89 | \( 1 - 16iT - 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.902337301472496733139408104652, −8.646963565966983029106201930584, −7.913312280305588642713215425180, −6.97665694776657729129051438115, −5.92760230442639695388624479270, −5.53423337621511941801479793110, −4.29625288946228132058118414308, −3.43042467485259959085456193143, −2.72835092344031485649477893492, −0.76393714206530891578627559963,
1.32689956029820022363625464688, 2.66129910792525779657171666232, 3.88548757733506825570768597457, 4.74299492762490571545970974386, 5.42650341598614599145451380584, 6.03944081049483700045557467413, 7.22676304236838813091087026639, 8.311602364099174959841546770019, 9.166898453577486070199159032513, 9.402095747839409739074834278817
