| L(s) = 1 | − 2-s + 4-s + 1.41·5-s − 8-s − 1.41·10-s − 4·11-s − 4.24·13-s + 16-s − 7.07·17-s + 5.65·19-s + 1.41·20-s + 4·22-s − 8·23-s − 2.99·25-s + 4.24·26-s − 2·29-s − 32-s + 7.07·34-s + 4·37-s − 5.65·38-s − 1.41·40-s + 9.89·41-s − 4·43-s − 4·44-s + 8·46-s + 5.65·47-s + 2.99·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.632·5-s − 0.353·8-s − 0.447·10-s − 1.20·11-s − 1.17·13-s + 0.250·16-s − 1.71·17-s + 1.29·19-s + 0.316·20-s + 0.852·22-s − 1.66·23-s − 0.599·25-s + 0.832·26-s − 0.371·29-s − 0.176·32-s + 1.21·34-s + 0.657·37-s − 0.917·38-s − 0.223·40-s + 1.54·41-s − 0.609·43-s − 0.603·44-s + 1.17·46-s + 0.825·47-s + 0.424·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 + 4.24T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 1.41T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 15.5T + 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 + 7.07T + 97T^{2} \) |
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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
−9.702980331019702016376976773919, −9.089529258812885745432561019029, −7.88266801611853841509255068199, −7.45395088274770400723346600429, −6.27160099821200432818040584398, −5.47310614112687396637632193631, −4.39436654013384319996796441894, −2.74798882208999716706009209848, −1.99225481395023532694876004319, 0,
1.99225481395023532694876004319, 2.74798882208999716706009209848, 4.39436654013384319996796441894, 5.47310614112687396637632193631, 6.27160099821200432818040584398, 7.45395088274770400723346600429, 7.88266801611853841509255068199, 9.089529258812885745432561019029, 9.702980331019702016376976773919
