| L(s) = 1 | + 3-s + 5-s + 7-s + 9-s + 2.47·11-s + 4.47·13-s + 15-s − 2·17-s − 6.47·19-s + 21-s + 4·23-s + 25-s + 27-s + 2·29-s + 10.4·31-s + 2.47·33-s + 35-s − 10.9·37-s + 4.47·39-s − 2·41-s + 8.94·43-s + 45-s − 4.94·47-s + 49-s − 2·51-s + 12.4·53-s + 2.47·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 0.377·7-s + 0.333·9-s + 0.745·11-s + 1.24·13-s + 0.258·15-s − 0.485·17-s − 1.48·19-s + 0.218·21-s + 0.834·23-s + 0.200·25-s + 0.192·27-s + 0.371·29-s + 1.88·31-s + 0.430·33-s + 0.169·35-s − 1.79·37-s + 0.716·39-s − 0.312·41-s + 1.36·43-s + 0.149·45-s − 0.721·47-s + 0.142·49-s − 0.280·51-s + 1.71·53-s + 0.333·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.384775508\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.384775508\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 2.47T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 10.9T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 8.94T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 + 4.94T + 79T^{2} \) |
| 83 | \( 1 + 0.944T + 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 + 0.472T + 97T^{2} \) |
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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
−8.364902093053644932051406839899, −7.18037873082995136544169424999, −6.55201247962866836090567648891, −6.08114258920083230552469031112, −5.02821552595597310209465790014, −4.29532660636841669767135234478, −3.64051279971926229872619421474, −2.66894362295743038344529145330, −1.84012667630769013414963451620, −0.980168657212867414090331246545,
0.980168657212867414090331246545, 1.84012667630769013414963451620, 2.66894362295743038344529145330, 3.64051279971926229872619421474, 4.29532660636841669767135234478, 5.02821552595597310209465790014, 6.08114258920083230552469031112, 6.55201247962866836090567648891, 7.18037873082995136544169424999, 8.364902093053644932051406839899
