| L(s) = 1 | − i·2-s − 4-s − 1.73·5-s + (2 − 1.73i)7-s + i·8-s + 1.73i·10-s − 3i·11-s − 3.46i·13-s + (−1.73 − 2i)14-s + 16-s − 6.92·17-s − 1.73i·19-s + 1.73·20-s − 3·22-s − 3i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s − 0.774·5-s + (0.755 − 0.654i)7-s + 0.353i·8-s + 0.547i·10-s − 0.904i·11-s − 0.960i·13-s + (−0.462 − 0.534i)14-s + 0.250·16-s − 1.68·17-s − 0.397i·19-s + 0.387·20-s − 0.639·22-s − 0.625i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.336354 - 0.902178i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.336354 - 0.902178i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2 + 1.73i)T \) |
| good | 5 | \( 1 + 1.73T + 5T^{2} \) |
| 11 | \( 1 + 3iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 + 1.73iT - 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 7T + 37T^{2} \) |
| 41 | \( 1 - 12.1T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 12iT - 53T^{2} \) |
| 59 | \( 1 + 3.46T + 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 3iT - 71T^{2} \) |
| 73 | \( 1 - 3.46iT - 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 - 5.19T + 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.05352991001634789015774597396, −10.49208491166796264591504321122, −9.135122218821324976242156531872, −8.248020643847419198410929074308, −7.54394187372462970620798312318, −6.12390283099408262641506106817, −4.70629313770613856950238198729, −3.94386026353510527376359751277, −2.58436935133502546713499710304, −0.65742208539132196488344514468,
2.09103980985007047506957425150, 4.08282557117265537752476112023, 4.75680924392216707947067771664, 6.04563459151078904046672920376, 7.13760509092754352747438475199, 7.88175738467019999506348158549, 8.860577689329856263759497452619, 9.555270959883553525954986347738, 11.03691457841884819142619487537, 11.65829347179685834056416686401
