| L(s) = 1 | − 2·7-s + 13-s + 4·17-s + 2·19-s + 6·23-s + 4·31-s + 2·37-s + 6·41-s − 4·43-s + 4·47-s − 3·49-s − 10·53-s − 8·59-s + 6·61-s + 8·67-s − 16·73-s − 4·83-s − 6·89-s − 2·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 8·119-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.277·13-s + 0.970·17-s + 0.458·19-s + 1.25·23-s + 0.718·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 1.37·53-s − 1.04·59-s + 0.768·61-s + 0.977·67-s − 1.87·73-s − 0.439·83-s − 0.635·89-s − 0.209·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.733·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.16721420763256, −13.50010163109903, −13.03158550545201, −12.68872940010520, −12.17578232103014, −11.57376825476564, −11.18228512556717, −10.57764382558933, −10.04807776694041, −9.626028272488417, −9.168286903545263, −8.633852455245896, −7.984851107218415, −7.536040281725500, −6.984325049759180, −6.388198893309220, −5.989181540985871, −5.325937222426956, −4.833765297794668, −4.143458454807159, −3.459203451540437, −3.016190907209636, −2.515577830907767, −1.435679403950516, −0.9724808346535537, 0,
0.9724808346535537, 1.435679403950516, 2.515577830907767, 3.016190907209636, 3.459203451540437, 4.143458454807159, 4.833765297794668, 5.325937222426956, 5.989181540985871, 6.388198893309220, 6.984325049759180, 7.536040281725500, 7.984851107218415, 8.633852455245896, 9.168286903545263, 9.626028272488417, 10.04807776694041, 10.57764382558933, 11.18228512556717, 11.57376825476564, 12.17578232103014, 12.68872940010520, 13.03158550545201, 13.50010163109903, 14.16721420763256
