| L(s) = 1 | + 2-s − 3-s + 4-s − 3.63·5-s − 6-s + 4.65·7-s + 8-s + 9-s − 3.63·10-s + 2.84·11-s − 12-s + 6.34·13-s + 4.65·14-s + 3.63·15-s + 16-s + 17-s + 18-s + 6.34·19-s − 3.63·20-s − 4.65·21-s + 2.84·22-s + 4.68·23-s − 24-s + 8.19·25-s + 6.34·26-s − 27-s + 4.65·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.62·5-s − 0.408·6-s + 1.76·7-s + 0.353·8-s + 0.333·9-s − 1.14·10-s + 0.859·11-s − 0.288·12-s + 1.76·13-s + 1.24·14-s + 0.937·15-s + 0.250·16-s + 0.242·17-s + 0.235·18-s + 1.45·19-s − 0.812·20-s − 1.01·21-s + 0.607·22-s + 0.976·23-s − 0.204·24-s + 1.63·25-s + 1.24·26-s − 0.192·27-s + 0.880·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.123120654\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.123120654\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| good | 5 | \( 1 + 3.63T + 5T^{2} \) |
| 7 | \( 1 - 4.65T + 7T^{2} \) |
| 11 | \( 1 - 2.84T + 11T^{2} \) |
| 13 | \( 1 - 6.34T + 13T^{2} \) |
| 19 | \( 1 - 6.34T + 19T^{2} \) |
| 23 | \( 1 - 4.68T + 23T^{2} \) |
| 29 | \( 1 + 4.11T + 29T^{2} \) |
| 31 | \( 1 + 7.08T + 31T^{2} \) |
| 37 | \( 1 + 6.00T + 37T^{2} \) |
| 41 | \( 1 + 0.683T + 41T^{2} \) |
| 43 | \( 1 - 9.97T + 43T^{2} \) |
| 47 | \( 1 - 0.499T + 47T^{2} \) |
| 53 | \( 1 - 8.61T + 53T^{2} \) |
| 61 | \( 1 + 9.67T + 61T^{2} \) |
| 67 | \( 1 - 8.49T + 67T^{2} \) |
| 71 | \( 1 + 4.60T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 15.2T + 83T^{2} \) |
| 89 | \( 1 - 2.14T + 89T^{2} \) |
| 97 | \( 1 - 2.64T + 97T^{2} \) |
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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
−7.82243374420695925073678345104, −7.40596165267693359660271520547, −6.76951632914449917143232369260, −5.58853257604632819083188675330, −5.28571783260355409690946206944, −4.22938515909067889125057673398, −3.95963238247536497179398730100, −3.18292828456272113190785488644, −1.55532273585185139520778871310, −0.997419449669069769123660119227,
0.997419449669069769123660119227, 1.55532273585185139520778871310, 3.18292828456272113190785488644, 3.95963238247536497179398730100, 4.22938515909067889125057673398, 5.28571783260355409690946206944, 5.58853257604632819083188675330, 6.76951632914449917143232369260, 7.40596165267693359660271520547, 7.82243374420695925073678345104
