| L(s) = 1 | − 1.35·2-s − 2.73·3-s − 0.170·4-s + 5-s + 3.70·6-s + 2.99·7-s + 2.93·8-s + 4.50·9-s − 1.35·10-s + 11-s + 0.467·12-s + 3.23·13-s − 4.04·14-s − 2.73·15-s − 3.62·16-s − 1.23·17-s − 6.08·18-s + 19-s − 0.170·20-s − 8.19·21-s − 1.35·22-s + 6.39·23-s − 8.04·24-s + 25-s − 4.37·26-s − 4.11·27-s − 0.510·28-s + ⋯ |
| L(s) = 1 | − 0.956·2-s − 1.58·3-s − 0.0853·4-s + 0.447·5-s + 1.51·6-s + 1.13·7-s + 1.03·8-s + 1.50·9-s − 0.427·10-s + 0.301·11-s + 0.135·12-s + 0.897·13-s − 1.08·14-s − 0.707·15-s − 0.907·16-s − 0.299·17-s − 1.43·18-s + 0.229·19-s − 0.0381·20-s − 1.78·21-s − 0.288·22-s + 1.33·23-s − 1.64·24-s + 0.200·25-s − 0.858·26-s − 0.791·27-s − 0.0965·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6653514077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6653514077\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + 1.35T + 2T^{2} \) |
| 3 | \( 1 + 2.73T + 3T^{2} \) |
| 7 | \( 1 - 2.99T + 7T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 1.23T + 17T^{2} \) |
| 23 | \( 1 - 6.39T + 23T^{2} \) |
| 29 | \( 1 - 0.202T + 29T^{2} \) |
| 31 | \( 1 + 5.80T + 31T^{2} \) |
| 37 | \( 1 + 8.57T + 37T^{2} \) |
| 41 | \( 1 + 8.14T + 41T^{2} \) |
| 43 | \( 1 - 3.02T + 43T^{2} \) |
| 47 | \( 1 - 6.80T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 1.12T + 61T^{2} \) |
| 67 | \( 1 - 4.71T + 67T^{2} \) |
| 71 | \( 1 + 7.63T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 2.93T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 - 7.44T + 89T^{2} \) |
| 97 | \( 1 + 4.12T + 97T^{2} \) |
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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
−10.12248814116185496185308596709, −9.014895503527976654450917971700, −8.511839836559878813741723341469, −7.32001898279167985280392873851, −6.67825731221817757828482940445, −5.44896978327888380410675780361, −5.06177974469747152338943614068, −3.97479649097216991407901367046, −1.74126054179276163342148334236, −0.856754334847730920024954892223,
0.856754334847730920024954892223, 1.74126054179276163342148334236, 3.97479649097216991407901367046, 5.06177974469747152338943614068, 5.44896978327888380410675780361, 6.67825731221817757828482940445, 7.32001898279167985280392873851, 8.511839836559878813741723341469, 9.014895503527976654450917971700, 10.12248814116185496185308596709
