| L(s) = 1 | + 11-s − 3·13-s − 4·17-s + 19-s − 3·23-s − 5·29-s + 3·31-s + 12·37-s − 8·41-s − 5·43-s + 8·47-s − 7·49-s − 10·53-s + 8·59-s + 10·61-s + 14·67-s − 5·71-s + 4·73-s + 8·79-s + 9·83-s − 3·89-s − 3·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.301·11-s − 0.832·13-s − 0.970·17-s + 0.229·19-s − 0.625·23-s − 0.928·29-s + 0.538·31-s + 1.97·37-s − 1.24·41-s − 0.762·43-s + 1.16·47-s − 49-s − 1.37·53-s + 1.04·59-s + 1.28·61-s + 1.71·67-s − 0.593·71-s + 0.468·73-s + 0.900·79-s + 0.987·83-s − 0.317·89-s − 0.304·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.455927265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.455927265\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−14.83821046243159, −14.34350691704140, −13.72173895920015, −13.24030143694796, −12.77023954411160, −12.16680766695761, −11.61437343111333, −11.22959803268170, −10.65055682433483, −9.831146147687322, −9.640379957614308, −9.056671825628499, −8.211054633109972, −7.989130191582991, −7.192896447080530, −6.625760573334419, −6.239422302203180, −5.323696575704098, −4.965701988139422, −4.136287735987223, −3.741038979190695, −2.750430205226728, −2.265345201481937, −1.464262699460449, −0.4367745029256930,
0.4367745029256930, 1.464262699460449, 2.265345201481937, 2.750430205226728, 3.741038979190695, 4.136287735987223, 4.965701988139422, 5.323696575704098, 6.239422302203180, 6.625760573334419, 7.192896447080530, 7.989130191582991, 8.211054633109972, 9.056671825628499, 9.640379957614308, 9.831146147687322, 10.65055682433483, 11.22959803268170, 11.61437343111333, 12.16680766695761, 12.77023954411160, 13.24030143694796, 13.72173895920015, 14.34350691704140, 14.83821046243159
