| L(s) = 1 | − 3·3-s − 3.13·7-s + 9·9-s − 54.7·11-s − 66.0·13-s − 47.6·17-s − 139.·19-s + 9.41·21-s + 100·23-s − 27·27-s − 75.4·29-s − 212.·31-s + 164.·33-s − 417.·37-s + 198.·39-s + 493.·41-s − 314.·43-s − 436.·47-s − 333.·49-s + 143.·51-s − 123.·53-s + 417.·57-s + 32.7·59-s + 573.·61-s − 28.2·63-s + 297.·67-s − 300·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.169·7-s + 0.333·9-s − 1.50·11-s − 1.40·13-s − 0.680·17-s − 1.68·19-s + 0.0978·21-s + 0.906·23-s − 0.192·27-s − 0.482·29-s − 1.22·31-s + 0.867·33-s − 1.85·37-s + 0.813·39-s + 1.87·41-s − 1.11·43-s − 1.35·47-s − 0.971·49-s + 0.392·51-s − 0.321·53-s + 0.970·57-s + 0.0721·59-s + 1.20·61-s − 0.0564·63-s + 0.541·67-s − 0.523·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1551069521\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1551069521\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 3.13T + 343T^{2} \) |
| 11 | \( 1 + 54.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 66.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 47.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 100T + 1.21e4T^{2} \) |
| 29 | \( 1 + 75.4T + 2.43e4T^{2} \) |
| 31 | \( 1 + 212.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 417.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 493.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 314.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 123.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 32.7T + 2.05e5T^{2} \) |
| 61 | \( 1 - 573.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 297.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 242.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 811.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 212.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 369.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 357.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 638.T + 9.12e5T^{2} \) |
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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
−8.574954770202019753282756004989, −7.76199439437792423825876010238, −7.01811938440979300709821568364, −6.37770360645166628829821595530, −5.14018674264831131621861663174, −5.03228123482184692964014369161, −3.83530838719734617270578552252, −2.63952992591585533454615630021, −1.91544162027109417728830259786, −0.16712655584304290244097772353,
0.16712655584304290244097772353, 1.91544162027109417728830259786, 2.63952992591585533454615630021, 3.83530838719734617270578552252, 5.03228123482184692964014369161, 5.14018674264831131621861663174, 6.37770360645166628829821595530, 7.01811938440979300709821568364, 7.76199439437792423825876010238, 8.574954770202019753282756004989
