| L(s) = 1 | + 3-s + 9-s + 11-s + 13-s + 2·17-s + 5·19-s + 6·23-s + 27-s − 2·29-s − 3·31-s + 33-s + 10·37-s + 39-s + 6·41-s − 3·43-s + 4·47-s + 2·51-s + 10·53-s + 5·57-s − 4·59-s − 5·61-s − 5·67-s + 6·69-s − 14·73-s + 4·79-s + 81-s − 6·83-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s + 0.301·11-s + 0.277·13-s + 0.485·17-s + 1.14·19-s + 1.25·23-s + 0.192·27-s − 0.371·29-s − 0.538·31-s + 0.174·33-s + 1.64·37-s + 0.160·39-s + 0.937·41-s − 0.457·43-s + 0.583·47-s + 0.280·51-s + 1.37·53-s + 0.662·57-s − 0.520·59-s − 0.640·61-s − 0.610·67-s + 0.722·69-s − 1.63·73-s + 0.450·79-s + 1/9·81-s − 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 323400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.716724860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.716724860\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 15 T + p T^{2} \) |
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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
−12.72018933043077, −12.04583104183566, −11.85959957630935, −11.09178829727571, −10.97598939631707, −10.24527555200336, −9.813216309797810, −9.376453046952630, −8.887218281347855, −8.713036155460819, −7.845205597463639, −7.517651419685468, −7.290096126045211, −6.577708424924836, −6.020116878709838, −5.615392685203464, −5.000990387105145, −4.534606208246604, −3.853680016452974, −3.533007004053603, −2.780182266842670, −2.605880093886909, −1.624765083811962, −1.181502470169710, −0.5928759723059036,
0.5928759723059036, 1.181502470169710, 1.624765083811962, 2.605880093886909, 2.780182266842670, 3.533007004053603, 3.853680016452974, 4.534606208246604, 5.000990387105145, 5.615392685203464, 6.020116878709838, 6.577708424924836, 7.290096126045211, 7.517651419685468, 7.845205597463639, 8.713036155460819, 8.887218281347855, 9.376453046952630, 9.813216309797810, 10.24527555200336, 10.97598939631707, 11.09178829727571, 11.85959957630935, 12.04583104183566, 12.72018933043077
