| L(s) = 1 | + 2.02·2-s + 3.01·3-s + 2.12·4-s + 6.11·6-s − 3.60·7-s + 0.245·8-s + 6.06·9-s + 5.37·11-s + 6.38·12-s + 0.621·13-s − 7.32·14-s − 3.74·16-s − 0.427·17-s + 12.3·18-s − 2.18·19-s − 10.8·21-s + 10.9·22-s − 6.22·23-s + 0.738·24-s + 1.26·26-s + 9.24·27-s − 7.64·28-s − 2.11·29-s − 31-s − 8.09·32-s + 16.1·33-s − 0.867·34-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.73·3-s + 1.06·4-s + 2.49·6-s − 1.36·7-s + 0.0867·8-s + 2.02·9-s + 1.62·11-s + 1.84·12-s + 0.172·13-s − 1.95·14-s − 0.935·16-s − 0.103·17-s + 2.90·18-s − 0.501·19-s − 2.37·21-s + 2.32·22-s − 1.29·23-s + 0.150·24-s + 0.247·26-s + 1.77·27-s − 1.44·28-s − 0.392·29-s − 0.179·31-s − 1.43·32-s + 2.81·33-s − 0.148·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.955049639\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.955049639\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 2.02T + 2T^{2} \) |
| 3 | \( 1 - 3.01T + 3T^{2} \) |
| 7 | \( 1 + 3.60T + 7T^{2} \) |
| 11 | \( 1 - 5.37T + 11T^{2} \) |
| 13 | \( 1 - 0.621T + 13T^{2} \) |
| 17 | \( 1 + 0.427T + 17T^{2} \) |
| 19 | \( 1 + 2.18T + 19T^{2} \) |
| 23 | \( 1 + 6.22T + 23T^{2} \) |
| 29 | \( 1 + 2.11T + 29T^{2} \) |
| 37 | \( 1 + 3.14T + 37T^{2} \) |
| 41 | \( 1 - 7.31T + 41T^{2} \) |
| 43 | \( 1 - 6.81T + 43T^{2} \) |
| 47 | \( 1 + 1.60T + 47T^{2} \) |
| 53 | \( 1 + 6.94T + 53T^{2} \) |
| 59 | \( 1 + 8.71T + 59T^{2} \) |
| 61 | \( 1 - 1.35T + 61T^{2} \) |
| 67 | \( 1 - 9.79T + 67T^{2} \) |
| 71 | \( 1 + 4.44T + 71T^{2} \) |
| 73 | \( 1 + 1.82T + 73T^{2} \) |
| 79 | \( 1 - 0.867T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 1.50T + 97T^{2} \) |
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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
−10.06392901061598361927730863210, −9.255129683235248230748300747038, −8.863052610937500400471405153709, −7.58395108869298208701425993954, −6.59611646052111761905386871846, −6.04189702023288069261631130919, −4.31962492029749998772890342522, −3.77728450856502653797395272705, −3.11987994414762759485525943187, −2.03096518272028380986244670990,
2.03096518272028380986244670990, 3.11987994414762759485525943187, 3.77728450856502653797395272705, 4.31962492029749998772890342522, 6.04189702023288069261631130919, 6.59611646052111761905386871846, 7.58395108869298208701425993954, 8.863052610937500400471405153709, 9.255129683235248230748300747038, 10.06392901061598361927730863210
