| L(s) = 1 | − 2.77·3-s + 4.69·7-s + 4.71·9-s − 6.40·11-s + 1.06·13-s + 1.91·17-s + 19-s − 13.0·21-s − 1.79·23-s − 4.75·27-s + 2.93·29-s + 5.55·31-s + 17.7·33-s − 11.4·37-s − 2.95·39-s − 1.14·41-s + 3.55·43-s − 10.8·47-s + 15.0·49-s − 5.32·51-s − 8.69·53-s − 2.77·57-s + 5.63·59-s − 3.39·61-s + 22.1·63-s + 8.82·67-s + 4.98·69-s + ⋯ |
| L(s) = 1 | − 1.60·3-s + 1.77·7-s + 1.57·9-s − 1.93·11-s + 0.295·13-s + 0.465·17-s + 0.229·19-s − 2.84·21-s − 0.374·23-s − 0.915·27-s + 0.545·29-s + 0.997·31-s + 3.09·33-s − 1.87·37-s − 0.473·39-s − 0.178·41-s + 0.542·43-s − 1.58·47-s + 2.14·49-s − 0.745·51-s − 1.19·53-s − 0.367·57-s + 0.733·59-s − 0.434·61-s + 2.78·63-s + 1.07·67-s + 0.600·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2.77T + 3T^{2} \) |
| 7 | \( 1 - 4.69T + 7T^{2} \) |
| 11 | \( 1 + 6.40T + 11T^{2} \) |
| 13 | \( 1 - 1.06T + 13T^{2} \) |
| 17 | \( 1 - 1.91T + 17T^{2} \) |
| 23 | \( 1 + 1.79T + 23T^{2} \) |
| 29 | \( 1 - 2.93T + 29T^{2} \) |
| 31 | \( 1 - 5.55T + 31T^{2} \) |
| 37 | \( 1 + 11.4T + 37T^{2} \) |
| 41 | \( 1 + 1.14T + 41T^{2} \) |
| 43 | \( 1 - 3.55T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 8.69T + 53T^{2} \) |
| 59 | \( 1 - 5.63T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 - 8.82T + 67T^{2} \) |
| 71 | \( 1 - 1.42T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 - 1.96T + 79T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.52694721425662980724354414992, −6.81588076599967759194689562028, −5.85811662851070170973867112904, −5.36874853555755854591952426657, −4.90013810607829543385536330002, −4.41471614223093769743007724274, −3.09241336888306745811816933900, −1.97337521072460976665496687686, −1.13708898012126983316434908206, 0,
1.13708898012126983316434908206, 1.97337521072460976665496687686, 3.09241336888306745811816933900, 4.41471614223093769743007724274, 4.90013810607829543385536330002, 5.36874853555755854591952426657, 5.85811662851070170973867112904, 6.81588076599967759194689562028, 7.52694721425662980724354414992
