| L(s) = 1 | − 3-s + 4.83·7-s + 9-s + 2.25·11-s + 13-s − 6.32·17-s − 2.58·19-s − 4.83·21-s − 4.83·23-s − 27-s + 9.09·29-s − 0.510·31-s − 2.25·33-s − 4.25·37-s − 39-s + 0.255·41-s − 9.16·43-s + 4.51·47-s + 16.4·49-s + 6.32·51-s + 9.93·53-s + 2.58·57-s + 14.1·59-s + 9.93·61-s + 4.83·63-s + 13.1·67-s + 4.83·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.82·7-s + 0.333·9-s + 0.679·11-s + 0.277·13-s − 1.53·17-s − 0.592·19-s − 1.05·21-s − 1.00·23-s − 0.192·27-s + 1.68·29-s − 0.0916·31-s − 0.392·33-s − 0.699·37-s − 0.160·39-s + 0.0398·41-s − 1.39·43-s + 0.657·47-s + 2.34·49-s + 0.886·51-s + 1.36·53-s + 0.342·57-s + 1.84·59-s + 1.27·61-s + 0.609·63-s + 1.60·67-s + 0.582·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.126550159\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.126550159\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4.83T + 7T^{2} \) |
| 11 | \( 1 - 2.25T + 11T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 + 4.83T + 23T^{2} \) |
| 29 | \( 1 - 9.09T + 29T^{2} \) |
| 31 | \( 1 + 0.510T + 31T^{2} \) |
| 37 | \( 1 + 4.25T + 37T^{2} \) |
| 41 | \( 1 - 0.255T + 41T^{2} \) |
| 43 | \( 1 + 9.16T + 43T^{2} \) |
| 47 | \( 1 - 4.51T + 47T^{2} \) |
| 53 | \( 1 - 9.93T + 53T^{2} \) |
| 59 | \( 1 - 14.1T + 59T^{2} \) |
| 61 | \( 1 - 9.93T + 61T^{2} \) |
| 67 | \( 1 - 13.1T + 67T^{2} \) |
| 71 | \( 1 + 5.74T + 71T^{2} \) |
| 73 | \( 1 + 2.90T + 73T^{2} \) |
| 79 | \( 1 - 5.74T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 - 3.74T + 89T^{2} \) |
| 97 | \( 1 - 12.5T + 97T^{2} \) |
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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
−8.011624286498535626592586537675, −6.93041136517787506798242368227, −6.63871240153221342085490185068, −5.66678247383174095845090383777, −5.01919880369671783130541656517, −4.34240483211921496672096916757, −3.89840924713107628819073531767, −2.35206918202470307670813960654, −1.77866166453167689708972984679, −0.77465680041494378385884869220,
0.77465680041494378385884869220, 1.77866166453167689708972984679, 2.35206918202470307670813960654, 3.89840924713107628819073531767, 4.34240483211921496672096916757, 5.01919880369671783130541656517, 5.66678247383174095845090383777, 6.63871240153221342085490185068, 6.93041136517787506798242368227, 8.011624286498535626592586537675
