| L(s) = 1 | + 3-s − 7-s + 9-s − 11-s − 13-s + 17-s + 19-s − 21-s + 23-s + 27-s + 29-s − 31-s − 33-s − 39-s + 41-s − 43-s − 47-s + 49-s + 51-s + 53-s + 57-s + 59-s + 61-s − 63-s + 67-s + 69-s + 71-s + ⋯ |
| L(s) = 1 | + 3-s − 7-s + 9-s − 11-s − 13-s + 17-s + 19-s − 21-s + 23-s + 27-s + 29-s − 31-s − 33-s − 39-s + 41-s − 43-s − 47-s + 49-s + 51-s + 53-s + 57-s + 59-s + 61-s − 63-s + 67-s + 69-s + 71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.968490656\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.968490656\) |
| \(L(1)\) |
\(\approx\) |
\(1.348163295\) |
| \(L(1)\) |
\(\approx\) |
\(1.348163295\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
| good | 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.597044878934877592344022715104, −19.82432264340175996615369024496, −19.27242929318491127413325967534, −18.596992281382950001199286519007, −17.86723211403599037252990429206, −16.65127924649530894434576803646, −16.102181655726558264935460066103, −15.34766332592801934993499208118, −14.58889106351705004008504604803, −13.89046088132977626793181711089, −12.93408911964553979815099178244, −12.66899382534337587483868959446, −11.57752933663650910841670425152, −10.24005501312088998266006188816, −9.927699293484388494676159659395, −9.146022499354062083976879229467, −8.21059444698359270110404980695, −7.39452902264587061698817397145, −6.88710095229229850050824509391, −5.56335888627716199056787440663, −4.82892374697454440439774692989, −3.57419746510904027057169045618, −2.992709775420941766491946683463, −2.27205810753055573815973483819, −0.88221143063316516468833070542,
0.88221143063316516468833070542, 2.27205810753055573815973483819, 2.992709775420941766491946683463, 3.57419746510904027057169045618, 4.82892374697454440439774692989, 5.56335888627716199056787440663, 6.88710095229229850050824509391, 7.39452902264587061698817397145, 8.21059444698359270110404980695, 9.146022499354062083976879229467, 9.927699293484388494676159659395, 10.24005501312088998266006188816, 11.57752933663650910841670425152, 12.66899382534337587483868959446, 12.93408911964553979815099178244, 13.89046088132977626793181711089, 14.58889106351705004008504604803, 15.34766332592801934993499208118, 16.102181655726558264935460066103, 16.65127924649530894434576803646, 17.86723211403599037252990429206, 18.596992281382950001199286519007, 19.27242929318491127413325967534, 19.82432264340175996615369024496, 20.597044878934877592344022715104
