| L(s) = 1 | + (1.71 − 1.03i)2-s + 1.73i·3-s + (1.86 − 3.53i)4-s + 2.98·5-s + (1.78 + 2.96i)6-s + (−0.449 − 7.98i)8-s − 2.99·9-s + (5.11 − 3.07i)10-s + 7.51i·11-s + (6.12 + 3.23i)12-s + 23.3·13-s + 5.16i·15-s + (−9.01 − 13.2i)16-s + 10.3·17-s + (−5.13 + 3.09i)18-s + 15.5i·19-s + ⋯ |
| L(s) = 1 | + (0.856 − 0.516i)2-s + 0.577i·3-s + (0.467 − 0.884i)4-s + 0.596·5-s + (0.297 + 0.494i)6-s + (−0.0562 − 0.998i)8-s − 0.333·9-s + (0.511 − 0.307i)10-s + 0.682i·11-s + (0.510 + 0.269i)12-s + 1.79·13-s + 0.344i·15-s + (−0.563 − 0.826i)16-s + 0.606·17-s + (−0.285 + 0.172i)18-s + 0.817i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 588 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.540373577\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.540373577\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.71 + 1.03i)T \) |
| 3 | \( 1 - 1.73iT \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 2.98T + 25T^{2} \) |
| 11 | \( 1 - 7.51iT - 121T^{2} \) |
| 13 | \( 1 - 23.3T + 169T^{2} \) |
| 17 | \( 1 - 10.3T + 289T^{2} \) |
| 19 | \( 1 - 15.5iT - 361T^{2} \) |
| 23 | \( 1 + 27.5iT - 529T^{2} \) |
| 29 | \( 1 - 19.6T + 841T^{2} \) |
| 31 | \( 1 + 28.5iT - 961T^{2} \) |
| 37 | \( 1 - 53.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 31.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 15.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 3.62iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 68.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 113. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 3.30T + 3.72e3T^{2} \) |
| 67 | \( 1 - 92.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 72.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 69.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 156. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 85.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 17.2T + 7.92e3T^{2} \) |
| 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
−10.43211365733999670047707163468, −9.878656878918156023865841114728, −8.947766434116225210030225206396, −7.74176845301016895800714218433, −6.16668749306662299381129830214, −5.94779761557015013386536822364, −4.59009439681097180306451105443, −3.83943986278499248510932771633, −2.63030086487462095809695444154, −1.29822710735067556622142332491,
1.38388325047217486935007759987, 2.88329479893557656567327432204, 3.84380464936331172469613202680, 5.29726404088528688512078804555, 6.02660759647518574963472471529, 6.65823151044164649594360355235, 7.82975959598385930568186319368, 8.498941187539327367923790348585, 9.519373850835996490729885888017, 11.02211812678620735628384045736
