| L(s) = 1 | − 2.41·2-s − 2.36·3-s + 3.83·4-s + 2.87·5-s + 5.71·6-s − 2.09·7-s − 4.42·8-s + 2.60·9-s − 6.93·10-s + 0.389·11-s − 9.06·12-s − 3.45·13-s + 5.06·14-s − 6.79·15-s + 3.01·16-s − 5.11·17-s − 6.28·18-s + 6.44·19-s + 11.0·20-s + 4.96·21-s − 0.940·22-s − 1.32·23-s + 10.4·24-s + 3.25·25-s + 8.35·26-s + 0.944·27-s − 8.04·28-s + ⋯ |
| L(s) = 1 | − 1.70·2-s − 1.36·3-s + 1.91·4-s + 1.28·5-s + 2.33·6-s − 0.793·7-s − 1.56·8-s + 0.867·9-s − 2.19·10-s + 0.117·11-s − 2.61·12-s − 0.959·13-s + 1.35·14-s − 1.75·15-s + 0.754·16-s − 1.24·17-s − 1.48·18-s + 1.47·19-s + 2.46·20-s + 1.08·21-s − 0.200·22-s − 0.277·23-s + 2.13·24-s + 0.650·25-s + 1.63·26-s + 0.181·27-s − 1.51·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3735002049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3735002049\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 \) |
| good | 2 | \( 1 + 2.41T + 2T^{2} \) |
| 3 | \( 1 + 2.36T + 3T^{2} \) |
| 5 | \( 1 - 2.87T + 5T^{2} \) |
| 7 | \( 1 + 2.09T + 7T^{2} \) |
| 11 | \( 1 - 0.389T + 11T^{2} \) |
| 13 | \( 1 + 3.45T + 13T^{2} \) |
| 17 | \( 1 + 5.11T + 17T^{2} \) |
| 19 | \( 1 - 6.44T + 19T^{2} \) |
| 23 | \( 1 + 1.32T + 23T^{2} \) |
| 29 | \( 1 + 0.597T + 29T^{2} \) |
| 31 | \( 1 + 1.23T + 31T^{2} \) |
| 37 | \( 1 - 2.57T + 37T^{2} \) |
| 43 | \( 1 - 9.91T + 43T^{2} \) |
| 47 | \( 1 + 2.21T + 47T^{2} \) |
| 53 | \( 1 + 5.40T + 53T^{2} \) |
| 59 | \( 1 - 6.60T + 59T^{2} \) |
| 61 | \( 1 + 6.39T + 61T^{2} \) |
| 67 | \( 1 + 5.58T + 67T^{2} \) |
| 71 | \( 1 - 7.00T + 71T^{2} \) |
| 73 | \( 1 + 14.4T + 73T^{2} \) |
| 79 | \( 1 + 5.98T + 79T^{2} \) |
| 83 | \( 1 - 15.4T + 83T^{2} \) |
| 89 | \( 1 - 0.334T + 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−9.472278417414008826773336339613, −8.992221082890781225279312329478, −7.67293107556296638074054555920, −6.94104116211986307902599155368, −6.28732758913407744034766082598, −5.70526537992576649322284163637, −4.70432077413412802494478041668, −2.81787396004491787820306346636, −1.81024953015891146484989791169, −0.57471439737994361177870512253,
0.57471439737994361177870512253, 1.81024953015891146484989791169, 2.81787396004491787820306346636, 4.70432077413412802494478041668, 5.70526537992576649322284163637, 6.28732758913407744034766082598, 6.94104116211986307902599155368, 7.67293107556296638074054555920, 8.992221082890781225279312329478, 9.472278417414008826773336339613
