| L(s) = 1 | + 0.779·2-s − 2.98·3-s − 1.39·4-s − 3.49·5-s − 2.32·6-s − 1.06·7-s − 2.64·8-s + 5.88·9-s − 2.72·10-s + 4.14·12-s + 0.0563·13-s − 0.832·14-s + 10.4·15-s + 0.720·16-s + 4.53·17-s + 4.58·18-s + 19-s + 4.86·20-s + 3.18·21-s − 1.07·23-s + 7.88·24-s + 7.19·25-s + 0.0439·26-s − 8.59·27-s + 1.48·28-s − 0.299·29-s + 8.11·30-s + ⋯ |
| L(s) = 1 | + 0.551·2-s − 1.72·3-s − 0.695·4-s − 1.56·5-s − 0.948·6-s − 0.403·7-s − 0.935·8-s + 1.96·9-s − 0.861·10-s + 1.19·12-s + 0.0156·13-s − 0.222·14-s + 2.68·15-s + 0.180·16-s + 1.10·17-s + 1.08·18-s + 0.229·19-s + 1.08·20-s + 0.694·21-s − 0.225·23-s + 1.60·24-s + 1.43·25-s + 0.00861·26-s − 1.65·27-s + 0.280·28-s − 0.0556·29-s + 1.48·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2299 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.779T + 2T^{2} \) |
| 3 | \( 1 + 2.98T + 3T^{2} \) |
| 5 | \( 1 + 3.49T + 5T^{2} \) |
| 7 | \( 1 + 1.06T + 7T^{2} \) |
| 13 | \( 1 - 0.0563T + 13T^{2} \) |
| 17 | \( 1 - 4.53T + 17T^{2} \) |
| 23 | \( 1 + 1.07T + 23T^{2} \) |
| 29 | \( 1 + 0.299T + 29T^{2} \) |
| 31 | \( 1 - 9.18T + 31T^{2} \) |
| 37 | \( 1 - 4.50T + 37T^{2} \) |
| 41 | \( 1 + 12.0T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 + 1.14T + 59T^{2} \) |
| 61 | \( 1 - 8.09T + 61T^{2} \) |
| 67 | \( 1 - 11.2T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 11.1T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 + 0.183T + 89T^{2} \) |
| 97 | \( 1 - 5.66T + 97T^{2} \) |
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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
−8.357990268071569949079107405660, −7.86208346495810355181840336892, −6.74738390313842979189412367870, −6.25488996570357204208741145061, −5.13912610347083354789489475271, −4.85711363781180229278229145803, −3.87581577508900554131457254532, −3.28020646486238580088965473081, −0.929178432015594279525258105925, 0,
0.929178432015594279525258105925, 3.28020646486238580088965473081, 3.87581577508900554131457254532, 4.85711363781180229278229145803, 5.13912610347083354789489475271, 6.25488996570357204208741145061, 6.74738390313842979189412367870, 7.86208346495810355181840336892, 8.357990268071569949079107405660
