| L(s) = 1 | − 2.23·3-s + 1.23·5-s + 4.23·7-s + 2.00·9-s + 4.23·11-s − 5.47·13-s − 2.76·15-s + 2·17-s + 7.70·19-s − 9.47·21-s − 2.23·23-s − 3.47·25-s + 2.23·27-s − 2.76·29-s − 7.70·31-s − 9.47·33-s + 5.23·35-s − 4.47·37-s + 12.2·39-s − 5.23·41-s − 10.2·43-s + 2.47·45-s − 4.23·47-s + 10.9·49-s − 4.47·51-s − 0.472·53-s + 5.23·55-s + ⋯ |
| L(s) = 1 | − 1.29·3-s + 0.552·5-s + 1.60·7-s + 0.666·9-s + 1.27·11-s − 1.51·13-s − 0.713·15-s + 0.485·17-s + 1.76·19-s − 2.06·21-s − 0.466·23-s − 0.694·25-s + 0.430·27-s − 0.513·29-s − 1.38·31-s − 1.64·33-s + 0.885·35-s − 0.735·37-s + 1.95·39-s − 0.817·41-s − 1.56·43-s + 0.368·45-s − 0.617·47-s + 1.56·49-s − 0.626·51-s − 0.0648·53-s + 0.706·55-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 | \( 1 + 2.23T + 3T^{2} \) |
| 5 | \( 1 - 1.23T + 5T^{2} \) |
| 7 | \( 1 - 4.23T + 7T^{2} \) |
| 11 | \( 1 - 4.23T + 11T^{2} \) |
| 13 | \( 1 + 5.47T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 7.70T + 19T^{2} \) |
| 23 | \( 1 + 2.23T + 23T^{2} \) |
| 29 | \( 1 + 2.76T + 29T^{2} \) |
| 31 | \( 1 + 7.70T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 5.23T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 4.23T + 47T^{2} \) |
| 53 | \( 1 + 0.472T + 53T^{2} \) |
| 59 | \( 1 + 8.94T + 59T^{2} \) |
| 61 | \( 1 - 7.47T + 61T^{2} \) |
| 67 | \( 1 + 4.70T + 67T^{2} \) |
| 71 | \( 1 + 0.763T + 71T^{2} \) |
| 73 | \( 1 + 7.52T + 73T^{2} \) |
| 79 | \( 1 + 12.9T + 79T^{2} \) |
| 83 | \( 1 + 16.7T + 83T^{2} \) |
| 89 | \( 1 + 14.4T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−7.25445600366208856916394748756, −6.91832467020040870539121620436, −5.75961316903722887355739237620, −5.46276534061150849656010852242, −4.92347604667195376247806181486, −4.19767104223064373053467955791, −3.12137151109480175688563203292, −1.72915620013153467827217334746, −1.44469799375910412984329104637, 0,
1.44469799375910412984329104637, 1.72915620013153467827217334746, 3.12137151109480175688563203292, 4.19767104223064373053467955791, 4.92347604667195376247806181486, 5.46276534061150849656010852242, 5.75961316903722887355739237620, 6.91832467020040870539121620436, 7.25445600366208856916394748756
