| L(s) = 1 | + 5-s + 2.73·7-s + 11-s + 4.19·13-s − 4.73·17-s + 7.46·19-s − 6.92·23-s + 25-s − 6.92·29-s + 0.535·31-s + 2.73·35-s − 0.535·37-s + 6.92·41-s + 12.1·43-s + 9.46·47-s + 0.464·49-s + 6·53-s + 55-s + 0.928·59-s − 7.46·61-s + 4.19·65-s + 4·67-s − 6·71-s − 9.66·73-s + 2.73·77-s + 7.46·79-s + 4.73·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.03·7-s + 0.301·11-s + 1.16·13-s − 1.14·17-s + 1.71·19-s − 1.44·23-s + 0.200·25-s − 1.28·29-s + 0.0962·31-s + 0.461·35-s − 0.0881·37-s + 1.08·41-s + 1.85·43-s + 1.38·47-s + 0.0663·49-s + 0.824·53-s + 0.134·55-s + 0.120·59-s − 0.955·61-s + 0.520·65-s + 0.488·67-s − 0.712·71-s − 1.13·73-s + 0.311·77-s + 0.839·79-s + 0.519·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.914929594\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.914929594\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 2.73T + 7T^{2} \) |
| 13 | \( 1 - 4.19T + 13T^{2} \) |
| 17 | \( 1 + 4.73T + 17T^{2} \) |
| 19 | \( 1 - 7.46T + 19T^{2} \) |
| 23 | \( 1 + 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 - 0.535T + 31T^{2} \) |
| 37 | \( 1 + 0.535T + 37T^{2} \) |
| 41 | \( 1 - 6.92T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 - 9.46T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 - 0.928T + 59T^{2} \) |
| 61 | \( 1 + 7.46T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 9.66T + 73T^{2} \) |
| 79 | \( 1 - 7.46T + 79T^{2} \) |
| 83 | \( 1 - 4.73T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 4.92T + 97T^{2} \) |
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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
−7.58826511203377093397565487603, −7.48227088765239604165134628489, −6.21206877568958306846962330794, −5.87742225591742689517417588041, −5.09807497359579653096917642335, −4.22526183742155019989444017494, −3.69005809956761748539795761813, −2.50220891814361461179387807846, −1.75541822608941386360133822539, −0.899592233996106505788903115189,
0.899592233996106505788903115189, 1.75541822608941386360133822539, 2.50220891814361461179387807846, 3.69005809956761748539795761813, 4.22526183742155019989444017494, 5.09807497359579653096917642335, 5.87742225591742689517417588041, 6.21206877568958306846962330794, 7.48227088765239604165134628489, 7.58826511203377093397565487603
