| L(s) = 1 | + 1.12·2-s − 0.729·4-s − 0.505·5-s + 1.69·7-s − 3.07·8-s − 0.569·10-s − 5.13·11-s + 6.20·13-s + 1.90·14-s − 2.00·16-s − 1.74·17-s − 8.35·19-s + 0.368·20-s − 5.78·22-s − 5.54·23-s − 4.74·25-s + 6.98·26-s − 1.23·28-s + 3.87·29-s − 2.30·31-s + 3.89·32-s − 1.96·34-s − 0.854·35-s + 5.41·37-s − 9.41·38-s + 1.55·40-s + 4.76·41-s + ⋯ |
| L(s) = 1 | + 0.796·2-s − 0.364·4-s − 0.226·5-s + 0.639·7-s − 1.08·8-s − 0.180·10-s − 1.54·11-s + 1.71·13-s + 0.509·14-s − 0.501·16-s − 0.422·17-s − 1.91·19-s + 0.0824·20-s − 1.23·22-s − 1.15·23-s − 0.948·25-s + 1.37·26-s − 0.233·28-s + 0.719·29-s − 0.413·31-s + 0.687·32-s − 0.336·34-s − 0.144·35-s + 0.889·37-s − 1.52·38-s + 0.245·40-s + 0.744·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1341 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 - 1.12T + 2T^{2} \) |
| 5 | \( 1 + 0.505T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 + 5.13T + 11T^{2} \) |
| 13 | \( 1 - 6.20T + 13T^{2} \) |
| 17 | \( 1 + 1.74T + 17T^{2} \) |
| 19 | \( 1 + 8.35T + 19T^{2} \) |
| 23 | \( 1 + 5.54T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 + 2.30T + 31T^{2} \) |
| 37 | \( 1 - 5.41T + 37T^{2} \) |
| 41 | \( 1 - 4.76T + 41T^{2} \) |
| 43 | \( 1 - 0.819T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 11.5T + 53T^{2} \) |
| 59 | \( 1 + 8.44T + 59T^{2} \) |
| 61 | \( 1 - 1.58T + 61T^{2} \) |
| 67 | \( 1 + 8.15T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 6.12T + 73T^{2} \) |
| 79 | \( 1 + 12.2T + 79T^{2} \) |
| 83 | \( 1 - 7.55T + 83T^{2} \) |
| 89 | \( 1 - 11.3T + 89T^{2} \) |
| 97 | \( 1 - 4.42T + 97T^{2} \) |
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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
−9.040422526040776169030674004616, −8.219547524323846398133628892240, −7.925771801180460972985655840301, −6.26363258200617641218038945559, −5.93680845819983263354633998831, −4.73288794771427970300136960629, −4.23876224253392829580662849768, −3.21002816888257143747660649644, −1.97880507067073609196664529858, 0,
1.97880507067073609196664529858, 3.21002816888257143747660649644, 4.23876224253392829580662849768, 4.73288794771427970300136960629, 5.93680845819983263354633998831, 6.26363258200617641218038945559, 7.925771801180460972985655840301, 8.219547524323846398133628892240, 9.040422526040776169030674004616
