| L(s) = 1 | + 0.388·3-s − 4.11·5-s + 1.27·7-s − 2.84·9-s + 4.41·11-s − 1.34·13-s − 1.59·15-s − 4.93·17-s + 2.24·19-s + 0.494·21-s − 0.728·23-s + 11.8·25-s − 2.27·27-s + 1.92·29-s + 3.21·31-s + 1.71·33-s − 5.22·35-s − 2.36·37-s − 0.521·39-s + 6.26·41-s − 5.03·43-s + 11.7·45-s + 7.79·47-s − 5.38·49-s − 1.91·51-s + 3.15·53-s − 18.1·55-s + ⋯ |
| L(s) = 1 | + 0.224·3-s − 1.83·5-s + 0.480·7-s − 0.949·9-s + 1.33·11-s − 0.371·13-s − 0.412·15-s − 1.19·17-s + 0.515·19-s + 0.107·21-s − 0.151·23-s + 2.37·25-s − 0.437·27-s + 0.356·29-s + 0.577·31-s + 0.298·33-s − 0.883·35-s − 0.388·37-s − 0.0834·39-s + 0.978·41-s − 0.768·43-s + 1.74·45-s + 1.13·47-s − 0.768·49-s − 0.268·51-s + 0.433·53-s − 2.44·55-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 | \( 1 - 0.388T + 3T^{2} \) |
| 5 | \( 1 + 4.11T + 5T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 - 4.41T + 11T^{2} \) |
| 13 | \( 1 + 1.34T + 13T^{2} \) |
| 17 | \( 1 + 4.93T + 17T^{2} \) |
| 19 | \( 1 - 2.24T + 19T^{2} \) |
| 23 | \( 1 + 0.728T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 31 | \( 1 - 3.21T + 31T^{2} \) |
| 37 | \( 1 + 2.36T + 37T^{2} \) |
| 41 | \( 1 - 6.26T + 41T^{2} \) |
| 43 | \( 1 + 5.03T + 43T^{2} \) |
| 47 | \( 1 - 7.79T + 47T^{2} \) |
| 53 | \( 1 - 3.15T + 53T^{2} \) |
| 59 | \( 1 - 0.926T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 + 8.56T + 67T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 0.150T + 79T^{2} \) |
| 83 | \( 1 - 9.70T + 83T^{2} \) |
| 89 | \( 1 + 15.1T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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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
−7.60553614182798402889507485354, −6.89998279883196790083932630798, −6.29519278385052504763580939605, −5.20707705632321672535021816051, −4.45870535429484006552821090728, −3.95623703529425972200204866814, −3.23918100576557218280859828649, −2.38310876075750673377585583529, −1.07060313150808519126931472505, 0,
1.07060313150808519126931472505, 2.38310876075750673377585583529, 3.23918100576557218280859828649, 3.95623703529425972200204866814, 4.45870535429484006552821090728, 5.20707705632321672535021816051, 6.29519278385052504763580939605, 6.89998279883196790083932630798, 7.60553614182798402889507485354
