| L(s) = 1 | + 2-s + 4-s + 3·7-s + 8-s + 3·14-s + 16-s + 8·17-s + 8·19-s + 3·28-s − 2·29-s + 6·31-s + 32-s + 8·34-s + 8·38-s − 5·41-s − 43-s − 5·47-s + 2·49-s + 8·53-s + 3·56-s − 2·58-s + 10·59-s + 7·61-s + 6·62-s + 64-s + 7·67-s + 8·68-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.13·7-s + 0.353·8-s + 0.801·14-s + 1/4·16-s + 1.94·17-s + 1.83·19-s + 0.566·28-s − 0.371·29-s + 1.07·31-s + 0.176·32-s + 1.37·34-s + 1.29·38-s − 0.780·41-s − 0.152·43-s − 0.729·47-s + 2/7·49-s + 1.09·53-s + 0.400·56-s − 0.262·58-s + 1.30·59-s + 0.896·61-s + 0.762·62-s + 1/8·64-s + 0.855·67-s + 0.970·68-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.607569095\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.607569095\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−14.32704452603077, −13.89166991781214, −13.70382988657424, −12.84326554249069, −12.32155396884467, −11.90337673432098, −11.40848772046465, −11.15713713580240, −10.22730190787291, −9.887151395688026, −9.452946889599959, −8.358841544132730, −8.158750046421786, −7.604740710240114, −7.054073962517429, −6.465514671261695, −5.564719295821767, −5.216328024335408, −5.030342965637031, −3.977086535353469, −3.580773660737121, −2.907627137245828, −2.186710178186590, −1.293022497640661, −0.9137666782358182,
0.9137666782358182, 1.293022497640661, 2.186710178186590, 2.907627137245828, 3.580773660737121, 3.977086535353469, 5.030342965637031, 5.216328024335408, 5.564719295821767, 6.465514671261695, 7.054073962517429, 7.604740710240114, 8.158750046421786, 8.358841544132730, 9.452946889599959, 9.887151395688026, 10.22730190787291, 11.15713713580240, 11.40848772046465, 11.90337673432098, 12.32155396884467, 12.84326554249069, 13.70382988657424, 13.89166991781214, 14.32704452603077
