| L(s) = 1 | − 10·5-s + 49·7-s − 340·11-s − 294·13-s − 1.22e3·17-s − 2.43e3·19-s + 2.00e3·23-s − 3.02e3·25-s + 6.74e3·29-s − 8.85e3·31-s − 490·35-s + 9.18e3·37-s + 1.45e4·41-s − 8.10e3·43-s − 312·47-s + 2.40e3·49-s + 1.46e4·53-s + 3.40e3·55-s − 2.76e4·59-s + 3.43e4·61-s + 2.94e3·65-s − 1.23e4·67-s + 3.69e4·71-s − 6.17e4·73-s − 1.66e4·77-s + 6.47e4·79-s − 7.70e4·83-s + ⋯ |
| L(s) = 1 | − 0.178·5-s + 0.377·7-s − 0.847·11-s − 0.482·13-s − 1.02·17-s − 1.54·19-s + 0.788·23-s − 0.967·25-s + 1.48·29-s − 1.65·31-s − 0.0676·35-s + 1.10·37-s + 1.35·41-s − 0.668·43-s − 0.0206·47-s + 1/7·49-s + 0.715·53-s + 0.151·55-s − 1.03·59-s + 1.18·61-s + 0.0863·65-s − 0.335·67-s + 0.869·71-s − 1.35·73-s − 0.320·77-s + 1.16·79-s − 1.22·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.283948706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.283948706\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{2} T \) |
| good | 5 | \( 1 + 2 p T + p^{5} T^{2} \) |
| 11 | \( 1 + 340 T + p^{5} T^{2} \) |
| 13 | \( 1 + 294 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1226 T + p^{5} T^{2} \) |
| 19 | \( 1 + 128 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 2000 T + p^{5} T^{2} \) |
| 29 | \( 1 - 6746 T + p^{5} T^{2} \) |
| 31 | \( 1 + 8856 T + p^{5} T^{2} \) |
| 37 | \( 1 - 9182 T + p^{5} T^{2} \) |
| 41 | \( 1 - 14574 T + p^{5} T^{2} \) |
| 43 | \( 1 + 8108 T + p^{5} T^{2} \) |
| 47 | \( 1 + 312 T + p^{5} T^{2} \) |
| 53 | \( 1 - 14634 T + p^{5} T^{2} \) |
| 59 | \( 1 + 27656 T + p^{5} T^{2} \) |
| 61 | \( 1 - 34338 T + p^{5} T^{2} \) |
| 67 | \( 1 + 12316 T + p^{5} T^{2} \) |
| 71 | \( 1 - 520 p T + p^{5} T^{2} \) |
| 73 | \( 1 + 61718 T + p^{5} T^{2} \) |
| 79 | \( 1 - 64752 T + p^{5} T^{2} \) |
| 83 | \( 1 + 77056 T + p^{5} T^{2} \) |
| 89 | \( 1 - 8166 T + p^{5} T^{2} \) |
| 97 | \( 1 - 20650 T + p^{5} T^{2} \) |
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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
−9.129084552063941351843545219823, −8.390998719207132445598034014327, −7.61659959180261779031617059981, −6.75720654148120326759521435714, −5.79734911028868829429511381154, −4.78309668990670809706097811050, −4.10757620990714873536279163862, −2.72845301430736139272729943890, −1.95815316574250326082369139107, −0.47139862332024387119970607919,
0.47139862332024387119970607919, 1.95815316574250326082369139107, 2.72845301430736139272729943890, 4.10757620990714873536279163862, 4.78309668990670809706097811050, 5.79734911028868829429511381154, 6.75720654148120326759521435714, 7.61659959180261779031617059981, 8.390998719207132445598034014327, 9.129084552063941351843545219823
