| L(s) = 1 | − 3-s + 5-s + 4.47·7-s + 9-s − 2·11-s + 4.47·13-s − 15-s + 1.07·17-s − 4·19-s − 4.47·21-s + 1.07·23-s + 25-s − 27-s − 3.54·29-s + 31-s + 2·33-s + 4.47·35-s + 2.47·37-s − 4.47·39-s + 4·41-s − 2·43-s + 45-s + 8.02·47-s + 13.0·49-s − 1.07·51-s + 10.0·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.69·7-s + 0.333·9-s − 0.603·11-s + 1.24·13-s − 0.258·15-s + 0.260·17-s − 0.917·19-s − 0.976·21-s + 0.223·23-s + 0.200·25-s − 0.192·27-s − 0.658·29-s + 0.179·31-s + 0.348·33-s + 0.756·35-s + 0.406·37-s − 0.716·39-s + 0.624·41-s − 0.304·43-s + 0.149·45-s + 1.17·47-s + 1.86·49-s − 0.150·51-s + 1.37·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.446690785\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.446690785\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 1.07T + 17T^{2} \) |
| 19 | \( 1 + 4T + 19T^{2} \) |
| 23 | \( 1 - 1.07T + 23T^{2} \) |
| 29 | \( 1 + 3.54T + 29T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 8.02T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 + 9.54T + 59T^{2} \) |
| 61 | \( 1 + 0.146T + 61T^{2} \) |
| 67 | \( 1 - 4.32T + 67T^{2} \) |
| 71 | \( 1 + 12.4T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 + 9.87T + 79T^{2} \) |
| 83 | \( 1 - 2.92T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 1.85T + 97T^{2} \) |
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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
−7.84396432422914535137436783681, −7.32758770802568776382966703820, −6.29842229260020471658399797930, −5.77983408429290111430625420143, −5.12566136270418499630087410249, −4.48050853624095071993639395543, −3.73918691858171651161531748168, −2.45930143481227438312705851916, −1.70140029846707046849484947243, −0.864253499474762383485478138928,
0.864253499474762383485478138928, 1.70140029846707046849484947243, 2.45930143481227438312705851916, 3.73918691858171651161531748168, 4.48050853624095071993639395543, 5.12566136270418499630087410249, 5.77983408429290111430625420143, 6.29842229260020471658399797930, 7.32758770802568776382966703820, 7.84396432422914535137436783681
