| L(s) = 1 | − 2.02·2-s − 3-s + 2.11·4-s + 2.02·6-s + 0.505·7-s − 0.227·8-s + 9-s + 0.687·11-s − 2.11·12-s − 5.78·13-s − 1.02·14-s − 3.76·16-s − 4.74·17-s − 2.02·18-s + 4.23·19-s − 0.505·21-s − 1.39·22-s + 8.36·23-s + 0.227·24-s + 11.7·26-s − 27-s + 1.06·28-s + 4.88·29-s + 2.68·31-s + 8.08·32-s − 0.687·33-s + 9.61·34-s + ⋯ |
| L(s) = 1 | − 1.43·2-s − 0.577·3-s + 1.05·4-s + 0.827·6-s + 0.191·7-s − 0.0806·8-s + 0.333·9-s + 0.207·11-s − 0.609·12-s − 1.60·13-s − 0.274·14-s − 0.940·16-s − 1.15·17-s − 0.477·18-s + 0.972·19-s − 0.110·21-s − 0.297·22-s + 1.74·23-s + 0.0465·24-s + 2.30·26-s − 0.192·27-s + 0.201·28-s + 0.907·29-s + 0.482·31-s + 1.42·32-s − 0.119·33-s + 1.64·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1875 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5292714156\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5292714156\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + 2.02T + 2T^{2} \) |
| 7 | \( 1 - 0.505T + 7T^{2} \) |
| 11 | \( 1 - 0.687T + 11T^{2} \) |
| 13 | \( 1 + 5.78T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 - 8.36T + 23T^{2} \) |
| 29 | \( 1 - 4.88T + 29T^{2} \) |
| 31 | \( 1 - 2.68T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 - 0.144T + 41T^{2} \) |
| 43 | \( 1 - 5.94T + 43T^{2} \) |
| 47 | \( 1 + 6.10T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 6.96T + 59T^{2} \) |
| 61 | \( 1 + 3.98T + 61T^{2} \) |
| 67 | \( 1 + 1.31T + 67T^{2} \) |
| 71 | \( 1 + 3.79T + 71T^{2} \) |
| 73 | \( 1 + 10.9T + 73T^{2} \) |
| 79 | \( 1 + 1.89T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 - 15.0T + 89T^{2} \) |
| 97 | \( 1 - 1.17T + 97T^{2} \) |
| 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
−9.113307553255347789775993098849, −8.735333001557745381578810188204, −7.54083781958819136780915561854, −7.17746473915811767406343908597, −6.40849571519051637644151346534, −5.03258281114763557253581719190, −4.63578320481153763565583000844, −2.96680357022576370672699099746, −1.83332300066461520333716165140, −0.62956313414414826670431414809,
0.62956313414414826670431414809, 1.83332300066461520333716165140, 2.96680357022576370672699099746, 4.63578320481153763565583000844, 5.03258281114763557253581719190, 6.40849571519051637644151346534, 7.17746473915811767406343908597, 7.54083781958819136780915561854, 8.735333001557745381578810188204, 9.113307553255347789775993098849
