| L(s) = 1 | + 0.879·3-s − 3.76·5-s − 7-s − 2.22·9-s + 2.66·11-s − 3.31·15-s − 5.08·17-s + 7.20·19-s − 0.879·21-s + 4.56·23-s + 9.19·25-s − 4.59·27-s − 7.15·29-s − 4.83·31-s + 2.34·33-s + 3.76·35-s + 6.21·37-s + 11.8·41-s + 1.74·43-s + 8.38·45-s − 5.21·47-s + 49-s − 4.47·51-s + 6.77·53-s − 10.0·55-s + 6.33·57-s − 7.72·59-s + ⋯ |
| L(s) = 1 | + 0.507·3-s − 1.68·5-s − 0.377·7-s − 0.742·9-s + 0.802·11-s − 0.855·15-s − 1.23·17-s + 1.65·19-s − 0.191·21-s + 0.952·23-s + 1.83·25-s − 0.884·27-s − 1.32·29-s − 0.868·31-s + 0.407·33-s + 0.636·35-s + 1.02·37-s + 1.85·41-s + 0.265·43-s + 1.25·45-s − 0.760·47-s + 0.142·49-s − 0.625·51-s + 0.930·53-s − 1.35·55-s + 0.839·57-s − 1.00·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.879T + 3T^{2} \) |
| 5 | \( 1 + 3.76T + 5T^{2} \) |
| 11 | \( 1 - 2.66T + 11T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 - 7.20T + 19T^{2} \) |
| 23 | \( 1 - 4.56T + 23T^{2} \) |
| 29 | \( 1 + 7.15T + 29T^{2} \) |
| 31 | \( 1 + 4.83T + 31T^{2} \) |
| 37 | \( 1 - 6.21T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 1.74T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 6.77T + 53T^{2} \) |
| 59 | \( 1 + 7.72T + 59T^{2} \) |
| 61 | \( 1 - 2.24T + 61T^{2} \) |
| 67 | \( 1 - 2.07T + 67T^{2} \) |
| 71 | \( 1 - 9.67T + 71T^{2} \) |
| 73 | \( 1 + 8.61T + 73T^{2} \) |
| 79 | \( 1 + 10.6T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 + 0.0838T + 89T^{2} \) |
| 97 | \( 1 + 11.2T + 97T^{2} \) |
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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
−7.48951411246951596115992028525, −6.92455831164810589904549616679, −6.04406040002181653518949778717, −5.22370643739312923418123479303, −4.32457453218558910510565554276, −3.74090752493851256023418054765, −3.20611581551869044518743865660, −2.42642885067697438679897247833, −1.03425106507182906271803733765, 0,
1.03425106507182906271803733765, 2.42642885067697438679897247833, 3.20611581551869044518743865660, 3.74090752493851256023418054765, 4.32457453218558910510565554276, 5.22370643739312923418123479303, 6.04406040002181653518949778717, 6.92455831164810589904549616679, 7.48951411246951596115992028525
