| L(s) = 1 | − 0.286·3-s − 0.286·7-s − 2.91·9-s + 4.26·11-s − 3.20·13-s − 0.286·17-s − 19-s + 0.0820·21-s − 0.936·23-s + 1.69·27-s + 2.26·29-s − 4.18·31-s − 1.22·33-s − 8.67·37-s + 0.917·39-s + 1.08·41-s − 4.42·43-s + 0.759·47-s − 6.91·49-s + 0.0820·51-s − 4.42·53-s + 0.286·57-s − 4.70·59-s + 10.1·61-s + 0.835·63-s − 9.82·67-s + 0.268·69-s + ⋯ |
| L(s) = 1 | − 0.165·3-s − 0.108·7-s − 0.972·9-s + 1.28·11-s − 0.888·13-s − 0.0694·17-s − 0.229·19-s + 0.0179·21-s − 0.195·23-s + 0.326·27-s + 0.421·29-s − 0.751·31-s − 0.212·33-s − 1.42·37-s + 0.146·39-s + 0.168·41-s − 0.675·43-s + 0.110·47-s − 0.988·49-s + 0.0114·51-s − 0.608·53-s + 0.0379·57-s − 0.611·59-s + 1.30·61-s + 0.105·63-s − 1.20·67-s + 0.0322·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.286T + 3T^{2} \) |
| 7 | \( 1 + 0.286T + 7T^{2} \) |
| 11 | \( 1 - 4.26T + 11T^{2} \) |
| 13 | \( 1 + 3.20T + 13T^{2} \) |
| 17 | \( 1 + 0.286T + 17T^{2} \) |
| 23 | \( 1 + 0.936T + 23T^{2} \) |
| 29 | \( 1 - 2.26T + 29T^{2} \) |
| 31 | \( 1 + 4.18T + 31T^{2} \) |
| 37 | \( 1 + 8.67T + 37T^{2} \) |
| 41 | \( 1 - 1.08T + 41T^{2} \) |
| 43 | \( 1 + 4.42T + 43T^{2} \) |
| 47 | \( 1 - 0.759T + 47T^{2} \) |
| 53 | \( 1 + 4.42T + 53T^{2} \) |
| 59 | \( 1 + 4.70T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 - 4.83T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 5.10T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 9.75T + 89T^{2} \) |
| 97 | \( 1 - 9.61T + 97T^{2} \) |
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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
−8.867526924795320193293153527014, −8.152371145571219894667841733405, −7.10835740305835903963143590385, −6.48194885747403117978648176427, −5.62727606246215731571720747450, −4.78208392998233713160815820697, −3.77681515280349990826022879062, −2.83998976776324888751182321096, −1.63498848647341875565445815630, 0,
1.63498848647341875565445815630, 2.83998976776324888751182321096, 3.77681515280349990826022879062, 4.78208392998233713160815820697, 5.62727606246215731571720747450, 6.48194885747403117978648176427, 7.10835740305835903963143590385, 8.152371145571219894667841733405, 8.867526924795320193293153527014
