| L(s) = 1 | + 5·5-s + 17.9·7-s − 39.5·11-s − 23.4·13-s + 16.2·17-s − 28.3·19-s + 105.·23-s + 25·25-s + 99.8·29-s − 321.·31-s + 89.5·35-s − 205.·37-s − 327.·41-s + 375.·43-s − 88.9·47-s − 22.4·49-s − 85.0·53-s − 197.·55-s − 796.·59-s − 317.·61-s − 117.·65-s − 525.·67-s + 760.·71-s − 101.·73-s − 708.·77-s − 225.·79-s + 503.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.966·7-s − 1.08·11-s − 0.501·13-s + 0.231·17-s − 0.342·19-s + 0.956·23-s + 0.200·25-s + 0.639·29-s − 1.86·31-s + 0.432·35-s − 0.914·37-s − 1.24·41-s + 1.33·43-s − 0.276·47-s − 0.0654·49-s − 0.220·53-s − 0.484·55-s − 1.75·59-s − 0.667·61-s − 0.224·65-s − 0.958·67-s + 1.27·71-s − 0.162·73-s − 1.04·77-s − 0.321·79-s + 0.665·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 5T \) |
| good | 7 | \( 1 - 17.9T + 343T^{2} \) |
| 11 | \( 1 + 39.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 23.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 16.2T + 4.91e3T^{2} \) |
| 19 | \( 1 + 28.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 105.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 99.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 321.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 205.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 327.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 375.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 88.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 85.0T + 1.48e5T^{2} \) |
| 59 | \( 1 + 796.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 317.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 525.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 760.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 101.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 225.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 503.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 868.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.74e3T + 9.12e5T^{2} \) |
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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
−8.660209235677717405343167868035, −7.76666133240971889193026549318, −7.21893461684100399783574212453, −6.10197189736479805679844154603, −5.14462134271069248662442360839, −4.78163000754598511651373460397, −3.38980006755331582032504976146, −2.35781905304776077197272595918, −1.45682185576788417667037168858, 0,
1.45682185576788417667037168858, 2.35781905304776077197272595918, 3.38980006755331582032504976146, 4.78163000754598511651373460397, 5.14462134271069248662442360839, 6.10197189736479805679844154603, 7.21893461684100399783574212453, 7.76666133240971889193026549318, 8.660209235677717405343167868035
