| L(s) = 1 | + 2.58·2-s + 4.66·4-s + 2.71·5-s − 4.30·7-s + 6.88·8-s + 7.01·10-s + 11-s + 13-s − 11.1·14-s + 8.43·16-s + 6.11·17-s − 0.300·19-s + 12.6·20-s + 2.58·22-s − 1.49·23-s + 2.39·25-s + 2.58·26-s − 20.0·28-s + 0.312·29-s + 4.55·31-s + 8.01·32-s + 15.7·34-s − 11.6·35-s − 7.96·37-s − 0.776·38-s + 18.7·40-s − 1.03·41-s + ⋯ |
| L(s) = 1 | + 1.82·2-s + 2.33·4-s + 1.21·5-s − 1.62·7-s + 2.43·8-s + 2.21·10-s + 0.301·11-s + 0.277·13-s − 2.96·14-s + 2.10·16-s + 1.48·17-s − 0.0689·19-s + 2.83·20-s + 0.550·22-s − 0.312·23-s + 0.478·25-s + 0.506·26-s − 3.79·28-s + 0.0579·29-s + 0.817·31-s + 1.41·32-s + 2.70·34-s − 1.97·35-s − 1.30·37-s − 0.125·38-s + 2.95·40-s − 0.160·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.459607332\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.459607332\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 - 2.58T + 2T^{2} \) |
| 5 | \( 1 - 2.71T + 5T^{2} \) |
| 7 | \( 1 + 4.30T + 7T^{2} \) |
| 17 | \( 1 - 6.11T + 17T^{2} \) |
| 19 | \( 1 + 0.300T + 19T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 - 0.312T + 29T^{2} \) |
| 31 | \( 1 - 4.55T + 31T^{2} \) |
| 37 | \( 1 + 7.96T + 37T^{2} \) |
| 41 | \( 1 + 1.03T + 41T^{2} \) |
| 43 | \( 1 - 12.4T + 43T^{2} \) |
| 47 | \( 1 + 13.1T + 47T^{2} \) |
| 53 | \( 1 + 8.49T + 53T^{2} \) |
| 59 | \( 1 + 9.22T + 59T^{2} \) |
| 61 | \( 1 + 9.86T + 61T^{2} \) |
| 67 | \( 1 + 8.05T + 67T^{2} \) |
| 71 | \( 1 - 0.696T + 71T^{2} \) |
| 73 | \( 1 - 4.57T + 73T^{2} \) |
| 79 | \( 1 + 17.0T + 79T^{2} \) |
| 83 | \( 1 + 8.76T + 83T^{2} \) |
| 89 | \( 1 - 3.24T + 89T^{2} \) |
| 97 | \( 1 - 7.63T + 97T^{2} \) |
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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
−9.940840129049833674909413408442, −9.095583923839325675742636287960, −7.60748152164832674544247473733, −6.63081313062869862491721702574, −6.07759608613665256143752955414, −5.66429397299995751080883477803, −4.56235233030427161033359065207, −3.39940367420306635127381455770, −2.95077797521043197148106635562, −1.66985287091851493744937062449,
1.66985287091851493744937062449, 2.95077797521043197148106635562, 3.39940367420306635127381455770, 4.56235233030427161033359065207, 5.66429397299995751080883477803, 6.07759608613665256143752955414, 6.63081313062869862491721702574, 7.60748152164832674544247473733, 9.095583923839325675742636287960, 9.940840129049833674909413408442
