| L(s) = 1 | + 0.263·5-s + 33.9·7-s − 38.5·11-s + 89.8·13-s + 107.·17-s − 39.7·19-s − 23·23-s − 124.·25-s + 261.·29-s + 279.·31-s + 8.93·35-s − 17.5·37-s − 37.7·41-s − 184.·43-s − 390.·47-s + 810.·49-s − 327.·53-s − 10.1·55-s + 529.·59-s + 129.·61-s + 23.6·65-s + 677.·67-s − 385.·71-s − 316.·73-s − 1.30e3·77-s − 1.32e3·79-s + 540.·83-s + ⋯ |
| L(s) = 1 | + 0.0235·5-s + 1.83·7-s − 1.05·11-s + 1.91·13-s + 1.53·17-s − 0.480·19-s − 0.208·23-s − 0.999·25-s + 1.67·29-s + 1.61·31-s + 0.0431·35-s − 0.0778·37-s − 0.143·41-s − 0.653·43-s − 1.21·47-s + 2.36·49-s − 0.849·53-s − 0.0248·55-s + 1.16·59-s + 0.272·61-s + 0.0451·65-s + 1.23·67-s − 0.644·71-s − 0.507·73-s − 1.93·77-s − 1.88·79-s + 0.715·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1656 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.250668168\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.250668168\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 0.263T + 125T^{2} \) |
| 7 | \( 1 - 33.9T + 343T^{2} \) |
| 11 | \( 1 + 38.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 89.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 39.7T + 6.85e3T^{2} \) |
| 29 | \( 1 - 261.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 279.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 17.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 37.7T + 6.89e4T^{2} \) |
| 43 | \( 1 + 184.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 390.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 327.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 529.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 129.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 677.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 385.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 316.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.32e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 540.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 661.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 599.T + 9.12e5T^{2} \) |
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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
−8.594623435111208727786366215160, −8.191069088120407536236628555052, −7.81285495895473453191012400240, −6.49846633366686236823361124029, −5.64801116663987844424854316213, −4.93486939266484970263934056331, −4.06971874897727534035135576364, −2.95370940160502227345677869374, −1.71732638660210705074959356240, −0.949494848178672257015388920948,
0.949494848178672257015388920948, 1.71732638660210705074959356240, 2.95370940160502227345677869374, 4.06971874897727534035135576364, 4.93486939266484970263934056331, 5.64801116663987844424854316213, 6.49846633366686236823361124029, 7.81285495895473453191012400240, 8.191069088120407536236628555052, 8.594623435111208727786366215160
