| L(s) = 1 | − 0.264·3-s + 1.70·5-s − 2.77·7-s − 2.93·9-s + 0.489·11-s + 0.503·13-s − 0.448·15-s + 3.96·17-s − 5.91·19-s + 0.732·21-s + 4.83·23-s − 2.10·25-s + 1.56·27-s + 3.23·29-s + 1.77·31-s − 0.129·33-s − 4.71·35-s + 2.45·37-s − 0.132·39-s + 1.21·41-s + 10.2·43-s − 4.98·45-s − 9.83·47-s + 0.691·49-s − 1.04·51-s − 1.49·53-s + 0.831·55-s + ⋯ |
| L(s) = 1 | − 0.152·3-s + 0.760·5-s − 1.04·7-s − 0.976·9-s + 0.147·11-s + 0.139·13-s − 0.115·15-s + 0.962·17-s − 1.35·19-s + 0.159·21-s + 1.00·23-s − 0.421·25-s + 0.301·27-s + 0.601·29-s + 0.318·31-s − 0.0224·33-s − 0.797·35-s + 0.403·37-s − 0.0212·39-s + 0.189·41-s + 1.55·43-s − 0.742·45-s − 1.43·47-s + 0.0988·49-s − 0.146·51-s − 0.205·53-s + 0.112·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 + 0.264T + 3T^{2} \) |
| 5 | \( 1 - 1.70T + 5T^{2} \) |
| 7 | \( 1 + 2.77T + 7T^{2} \) |
| 11 | \( 1 - 0.489T + 11T^{2} \) |
| 13 | \( 1 - 0.503T + 13T^{2} \) |
| 17 | \( 1 - 3.96T + 17T^{2} \) |
| 19 | \( 1 + 5.91T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 - 3.23T + 29T^{2} \) |
| 31 | \( 1 - 1.77T + 31T^{2} \) |
| 37 | \( 1 - 2.45T + 37T^{2} \) |
| 41 | \( 1 - 1.21T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 9.83T + 47T^{2} \) |
| 53 | \( 1 + 1.49T + 53T^{2} \) |
| 59 | \( 1 - 3.72T + 59T^{2} \) |
| 61 | \( 1 + 7.69T + 61T^{2} \) |
| 67 | \( 1 - 2.43T + 67T^{2} \) |
| 71 | \( 1 - 0.608T + 71T^{2} \) |
| 73 | \( 1 + 2.67T + 73T^{2} \) |
| 79 | \( 1 + 4.84T + 79T^{2} \) |
| 83 | \( 1 + 8.43T + 83T^{2} \) |
| 89 | \( 1 + 0.160T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−7.44760939002393018051685348351, −6.44881626341780229686725519378, −6.23177080717415297923004340185, −5.57285762226302738274533097217, −4.77289235744181464574060986609, −3.79134171319492414541196500584, −2.98427205952957399357416538157, −2.41045066385317163337635020398, −1.20218043659761289514704718627, 0,
1.20218043659761289514704718627, 2.41045066385317163337635020398, 2.98427205952957399357416538157, 3.79134171319492414541196500584, 4.77289235744181464574060986609, 5.57285762226302738274533097217, 6.23177080717415297923004340185, 6.44881626341780229686725519378, 7.44760939002393018051685348351
