L(s) = 1 | + 7-s + 2·17-s − 4·19-s + 4·23-s − 5·25-s − 6·29-s + 4·31-s + 6·37-s + 6·41-s − 4·43-s + 12·47-s + 49-s − 4·53-s + 8·61-s + 12·67-s − 4·71-s − 4·73-s − 4·83-s − 12·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·119-s + ⋯ |
L(s) = 1 | + 0.377·7-s + 0.485·17-s − 0.917·19-s + 0.834·23-s − 25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 1.75·47-s + 1/7·49-s − 0.549·53-s + 1.02·61-s + 1.46·67-s − 0.474·71-s − 0.468·73-s − 0.439·83-s − 1.27·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60984 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.342300473\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.342300473\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−14.24609994136875, −13.88383118455299, −13.22632815149997, −12.77143125848159, −12.38506062686958, −11.65535516663532, −11.21003370940075, −10.91326057140761, −10.09374624136274, −9.787038902315435, −9.139293411814249, −8.576877110448399, −8.121556682195905, −7.467894753669532, −7.134374682394668, −6.303129084779936, −5.845516132650509, −5.323766129799568, −4.581902239904535, −4.097999678396850, −3.505452063152600, −2.649205939763665, −2.146427615763255, −1.322429722065710, −0.5419168126348404,
0.5419168126348404, 1.322429722065710, 2.146427615763255, 2.649205939763665, 3.505452063152600, 4.097999678396850, 4.581902239904535, 5.323766129799568, 5.845516132650509, 6.303129084779936, 7.134374682394668, 7.467894753669532, 8.121556682195905, 8.576877110448399, 9.139293411814249, 9.787038902315435, 10.09374624136274, 10.91326057140761, 11.21003370940075, 11.65535516663532, 12.38506062686958, 12.77143125848159, 13.22632815149997, 13.88383118455299, 14.24609994136875