| L(s) = 1 | − 0.248·2-s + 2.33·3-s − 1.93·4-s − 1.48·5-s − 0.582·6-s + 2.77·7-s + 0.980·8-s + 2.47·9-s + 0.369·10-s + 11-s − 4.53·12-s − 4.90·13-s − 0.690·14-s − 3.47·15-s + 3.63·16-s + 4.14·17-s − 0.615·18-s − 3.68·19-s + 2.87·20-s + 6.48·21-s − 0.248·22-s + 0.319·23-s + 2.29·24-s − 2.79·25-s + 1.22·26-s − 1.23·27-s − 5.37·28-s + ⋯ |
| L(s) = 1 | − 0.176·2-s + 1.35·3-s − 0.969·4-s − 0.664·5-s − 0.237·6-s + 1.04·7-s + 0.346·8-s + 0.824·9-s + 0.116·10-s + 0.301·11-s − 1.30·12-s − 1.35·13-s − 0.184·14-s − 0.896·15-s + 0.907·16-s + 1.00·17-s − 0.145·18-s − 0.846·19-s + 0.643·20-s + 1.41·21-s − 0.0530·22-s + 0.0666·23-s + 0.468·24-s − 0.558·25-s + 0.239·26-s − 0.237·27-s − 1.01·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9251 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 + 0.248T + 2T^{2} \) |
| 3 | \( 1 - 2.33T + 3T^{2} \) |
| 5 | \( 1 + 1.48T + 5T^{2} \) |
| 7 | \( 1 - 2.77T + 7T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 - 4.14T + 17T^{2} \) |
| 19 | \( 1 + 3.68T + 19T^{2} \) |
| 23 | \( 1 - 0.319T + 23T^{2} \) |
| 31 | \( 1 + 5.30T + 31T^{2} \) |
| 37 | \( 1 - 7.08T + 37T^{2} \) |
| 41 | \( 1 + 8.26T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 - 1.30T + 61T^{2} \) |
| 67 | \( 1 - 5.40T + 67T^{2} \) |
| 71 | \( 1 + 8.32T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 6.15T + 79T^{2} \) |
| 83 | \( 1 - 9.25T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 5.67T + 97T^{2} \) |
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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
−7.58164487000636617417150470700, −7.25923262325014534837019326120, −5.82850072200326561799514358733, −5.13022447233354073277055144045, −4.27893650326414111322769278376, −4.00624012384006710060307016431, −3.07307763085861748636552534300, −2.23383406640175783229355172660, −1.32756142653248705236580757209, 0,
1.32756142653248705236580757209, 2.23383406640175783229355172660, 3.07307763085861748636552534300, 4.00624012384006710060307016431, 4.27893650326414111322769278376, 5.13022447233354073277055144045, 5.82850072200326561799514358733, 7.25923262325014534837019326120, 7.58164487000636617417150470700
