| L(s) = 1 | + 0.106·2-s − 3.24·3-s − 1.98·4-s − 2.21·5-s − 0.347·6-s − 0.711·7-s − 0.426·8-s + 7.55·9-s − 0.236·10-s − 0.120·11-s + 6.46·12-s − 13-s − 0.0760·14-s + 7.20·15-s + 3.93·16-s + 3.52·17-s + 0.807·18-s − 3.48·19-s + 4.40·20-s + 2.31·21-s − 0.0129·22-s + 7.43·23-s + 1.38·24-s − 0.0846·25-s − 0.106·26-s − 14.7·27-s + 1.41·28-s + ⋯ |
| L(s) = 1 | + 0.0755·2-s − 1.87·3-s − 0.994·4-s − 0.991·5-s − 0.141·6-s − 0.268·7-s − 0.150·8-s + 2.51·9-s − 0.0749·10-s − 0.0364·11-s + 1.86·12-s − 0.277·13-s − 0.0203·14-s + 1.85·15-s + 0.982·16-s + 0.854·17-s + 0.190·18-s − 0.798·19-s + 0.985·20-s + 0.504·21-s − 0.00275·22-s + 1.54·23-s + 0.282·24-s − 0.0169·25-s − 0.0209·26-s − 2.84·27-s + 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 - 0.106T + 2T^{2} \) |
| 3 | \( 1 + 3.24T + 3T^{2} \) |
| 5 | \( 1 + 2.21T + 5T^{2} \) |
| 7 | \( 1 + 0.711T + 7T^{2} \) |
| 11 | \( 1 + 0.120T + 11T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 + 3.48T + 19T^{2} \) |
| 23 | \( 1 - 7.43T + 23T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 - 6.35T + 31T^{2} \) |
| 37 | \( 1 + 10.2T + 37T^{2} \) |
| 41 | \( 1 + 2.21T + 41T^{2} \) |
| 43 | \( 1 - 8.37T + 43T^{2} \) |
| 47 | \( 1 - 4.98T + 47T^{2} \) |
| 53 | \( 1 + 5.63T + 53T^{2} \) |
| 59 | \( 1 - 0.390T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 15.4T + 67T^{2} \) |
| 71 | \( 1 + 10.5T + 71T^{2} \) |
| 73 | \( 1 - 6.51T + 73T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 + 8.01T + 83T^{2} \) |
| 89 | \( 1 + 0.153T + 89T^{2} \) |
| 97 | \( 1 - 8.74T + 97T^{2} \) |
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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
−9.387887323132449291880539820827, −8.334616777790702705014471667290, −7.43592923704664985914906656502, −6.63056301581828054455777068275, −5.73842802265570556735531255729, −4.90283932913061455767091139635, −4.40380841245493005008424180080, −3.38114327666371683693492810986, −1.03328045117068318039278855394, 0,
1.03328045117068318039278855394, 3.38114327666371683693492810986, 4.40380841245493005008424180080, 4.90283932913061455767091139635, 5.73842802265570556735531255729, 6.63056301581828054455777068275, 7.43592923704664985914906656502, 8.334616777790702705014471667290, 9.387887323132449291880539820827
