| L(s) = 1 | − 3-s − 19·7-s − 26·9-s + 20·11-s + 77·13-s + 11·17-s − 19·19-s + 19·21-s − 79·23-s + 53·27-s − 303·29-s + 214·31-s − 20·33-s + 250·37-s − 77·39-s − 230·41-s + 402·43-s − 48·47-s + 18·49-s − 11·51-s + 417·53-s + 19·57-s + 99·59-s + 332·61-s + 494·63-s + 319·67-s + 79·69-s + ⋯ |
| L(s) = 1 | − 0.192·3-s − 1.02·7-s − 0.962·9-s + 0.548·11-s + 1.64·13-s + 0.156·17-s − 0.229·19-s + 0.197·21-s − 0.716·23-s + 0.377·27-s − 1.94·29-s + 1.23·31-s − 0.105·33-s + 1.11·37-s − 0.316·39-s − 0.876·41-s + 1.42·43-s − 0.148·47-s + 0.0524·49-s − 0.0302·51-s + 1.08·53-s + 0.0441·57-s + 0.218·59-s + 0.696·61-s + 0.987·63-s + 0.581·67-s + 0.137·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + p T \) |
| good | 3 | \( 1 + T + p^{3} T^{2} \) |
| 7 | \( 1 + 19 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 - 77 T + p^{3} T^{2} \) |
| 17 | \( 1 - 11 T + p^{3} T^{2} \) |
| 23 | \( 1 + 79 T + p^{3} T^{2} \) |
| 29 | \( 1 + 303 T + p^{3} T^{2} \) |
| 31 | \( 1 - 214 T + p^{3} T^{2} \) |
| 37 | \( 1 - 250 T + p^{3} T^{2} \) |
| 41 | \( 1 + 230 T + p^{3} T^{2} \) |
| 43 | \( 1 - 402 T + p^{3} T^{2} \) |
| 47 | \( 1 + 48 T + p^{3} T^{2} \) |
| 53 | \( 1 - 417 T + p^{3} T^{2} \) |
| 59 | \( 1 - 99 T + p^{3} T^{2} \) |
| 61 | \( 1 - 332 T + p^{3} T^{2} \) |
| 67 | \( 1 - 319 T + p^{3} T^{2} \) |
| 71 | \( 1 + 1088 T + p^{3} T^{2} \) |
| 73 | \( 1 - 373 T + p^{3} T^{2} \) |
| 79 | \( 1 - 102 T + p^{3} T^{2} \) |
| 83 | \( 1 + 934 T + p^{3} T^{2} \) |
| 89 | \( 1 - 498 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1386 T + p^{3} T^{2} \) |
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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
−8.585290763677679997234240973143, −7.75699141772626498252343873406, −6.61252432099887349126501757329, −6.10336212059521382223525639147, −5.52498664023931716424608757298, −4.08758087949949226868353638413, −3.51405996214655513134019488739, −2.49662233880183450845382116962, −1.13540099310160617403245917882, 0,
1.13540099310160617403245917882, 2.49662233880183450845382116962, 3.51405996214655513134019488739, 4.08758087949949226868353638413, 5.52498664023931716424608757298, 6.10336212059521382223525639147, 6.61252432099887349126501757329, 7.75699141772626498252343873406, 8.585290763677679997234240973143
