| L(s) = 1 | − 2.29·2-s − 3·3-s − 2.71·4-s − 16.8·5-s + 6.89·6-s − 2.53·7-s + 24.6·8-s + 9·9-s + 38.7·10-s + 11·11-s + 8.13·12-s − 6.61·13-s + 5.82·14-s + 50.5·15-s − 34.9·16-s − 115.·17-s − 20.6·18-s − 18.0·19-s + 45.7·20-s + 7.59·21-s − 25.2·22-s − 112.·23-s − 73.8·24-s + 159.·25-s + 15.2·26-s − 27·27-s + 6.86·28-s + ⋯ |
| L(s) = 1 | − 0.812·2-s − 0.577·3-s − 0.339·4-s − 1.50·5-s + 0.469·6-s − 0.136·7-s + 1.08·8-s + 0.333·9-s + 1.22·10-s + 0.301·11-s + 0.195·12-s − 0.141·13-s + 0.111·14-s + 0.870·15-s − 0.545·16-s − 1.65·17-s − 0.270·18-s − 0.217·19-s + 0.511·20-s + 0.0789·21-s − 0.245·22-s − 1.02·23-s − 0.628·24-s + 1.27·25-s + 0.114·26-s − 0.192·27-s + 0.0463·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 + 61T \) |
| good | 2 | \( 1 + 2.29T + 8T^{2} \) |
| 5 | \( 1 + 16.8T + 125T^{2} \) |
| 7 | \( 1 + 2.53T + 343T^{2} \) |
| 13 | \( 1 + 6.61T + 2.19e3T^{2} \) |
| 17 | \( 1 + 115.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 18.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 196.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 333.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 277.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 185.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 174.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 749.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 781.T + 2.05e5T^{2} \) |
| 67 | \( 1 - 604.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 763.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 624.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 489.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 714.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.61e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
| 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
−8.519288680402238732678383653964, −7.67950641319725601610469318428, −7.09456422964598367155690402074, −6.23746480839812316240965421566, −4.94766229125891460904293648444, −4.28425525960452169219548184771, −3.69222555502216933028062566511, −2.09104547599037861769750407489, −0.70697540140630557097956268296, 0,
0.70697540140630557097956268296, 2.09104547599037861769750407489, 3.69222555502216933028062566511, 4.28425525960452169219548184771, 4.94766229125891460904293648444, 6.23746480839812316240965421566, 7.09456422964598367155690402074, 7.67950641319725601610469318428, 8.519288680402238732678383653964
