L(s) = 1 | + 0.551·5-s + 7-s + 0.103·11-s + 5.69·13-s + 4.24·17-s + 19-s − 23-s − 4.69·25-s − 1.14·29-s + 6.24·31-s + 0.551·35-s + 3.35·37-s + 9.28·41-s − 0.799·43-s − 8.14·47-s + 49-s − 0.695·53-s + 0.0573·55-s + 0.695·59-s + 7.55·61-s + 3.14·65-s − 0.551·67-s − 12.1·71-s − 3.14·73-s + 0.103·77-s + 8.45·79-s + 1.14·83-s + ⋯ |
L(s) = 1 | + 0.246·5-s + 0.377·7-s + 0.0313·11-s + 1.57·13-s + 1.03·17-s + 0.229·19-s − 0.208·23-s − 0.939·25-s − 0.212·29-s + 1.12·31-s + 0.0932·35-s + 0.550·37-s + 1.45·41-s − 0.121·43-s − 1.18·47-s + 0.142·49-s − 0.0955·53-s + 0.00772·55-s + 0.0905·59-s + 0.966·61-s + 0.389·65-s − 0.0674·67-s − 1.44·71-s − 0.367·73-s + 0.0118·77-s + 0.951·79-s + 0.125·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5796 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.582622640\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.582622640\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 5 | \( 1 - 0.551T + 5T^{2} \) |
| 11 | \( 1 - 0.103T + 11T^{2} \) |
| 13 | \( 1 - 5.69T + 13T^{2} \) |
| 17 | \( 1 - 4.24T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 29 | \( 1 + 1.14T + 29T^{2} \) |
| 31 | \( 1 - 6.24T + 31T^{2} \) |
| 37 | \( 1 - 3.35T + 37T^{2} \) |
| 41 | \( 1 - 9.28T + 41T^{2} \) |
| 43 | \( 1 + 0.799T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 + 0.695T + 53T^{2} \) |
| 59 | \( 1 - 0.695T + 59T^{2} \) |
| 61 | \( 1 - 7.55T + 61T^{2} \) |
| 67 | \( 1 + 0.551T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + 3.14T + 73T^{2} \) |
| 79 | \( 1 - 8.45T + 79T^{2} \) |
| 83 | \( 1 - 1.14T + 83T^{2} \) |
| 89 | \( 1 + 9.69T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−8.048568438011909494591835183032, −7.60555777314351810374605809376, −6.52800708093508465447041980649, −5.97177105093556601731094545077, −5.38091053171070914288871033057, −4.37811672185752591495360307290, −3.69578728058189584835301744050, −2.84726938587457551678902952755, −1.72608131658736950012558364858, −0.916221098583978310017288835615,
0.916221098583978310017288835615, 1.72608131658736950012558364858, 2.84726938587457551678902952755, 3.69578728058189584835301744050, 4.37811672185752591495360307290, 5.38091053171070914288871033057, 5.97177105093556601731094545077, 6.52800708093508465447041980649, 7.60555777314351810374605809376, 8.048568438011909494591835183032