| L(s) = 1 | + 2-s − 3.03·3-s + 4-s − 2.36·5-s − 3.03·6-s − 0.910·7-s + 8-s + 6.20·9-s − 2.36·10-s − 2.58·11-s − 3.03·12-s + 1.05·13-s − 0.910·14-s + 7.16·15-s + 16-s − 6.94·17-s + 6.20·18-s + 19-s − 2.36·20-s + 2.76·21-s − 2.58·22-s − 2.67·23-s − 3.03·24-s + 0.582·25-s + 1.05·26-s − 9.72·27-s − 0.910·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.75·3-s + 0.5·4-s − 1.05·5-s − 1.23·6-s − 0.344·7-s + 0.353·8-s + 2.06·9-s − 0.747·10-s − 0.780·11-s − 0.875·12-s + 0.291·13-s − 0.243·14-s + 1.85·15-s + 0.250·16-s − 1.68·17-s + 1.46·18-s + 0.229·19-s − 0.528·20-s + 0.602·21-s − 0.551·22-s − 0.558·23-s − 0.619·24-s + 0.116·25-s + 0.205·26-s − 1.87·27-s − 0.172·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3588086445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3588086445\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 + 3.03T + 3T^{2} \) |
| 5 | \( 1 + 2.36T + 5T^{2} \) |
| 7 | \( 1 + 0.910T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 + 6.94T + 17T^{2} \) |
| 23 | \( 1 + 2.67T + 23T^{2} \) |
| 29 | \( 1 + 5.00T + 29T^{2} \) |
| 31 | \( 1 + 4.06T + 31T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 - 7.52T + 41T^{2} \) |
| 43 | \( 1 + 4.17T + 43T^{2} \) |
| 47 | \( 1 + 3.63T + 47T^{2} \) |
| 53 | \( 1 + 9.88T + 53T^{2} \) |
| 59 | \( 1 + 3.88T + 59T^{2} \) |
| 61 | \( 1 + 4.39T + 61T^{2} \) |
| 67 | \( 1 - 2.21T + 67T^{2} \) |
| 71 | \( 1 + 8.42T + 71T^{2} \) |
| 73 | \( 1 - 9.64T + 73T^{2} \) |
| 79 | \( 1 - 1.16T + 79T^{2} \) |
| 83 | \( 1 + 12.6T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 9.67T + 97T^{2} \) |
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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
−7.58535490667806779030229944011, −6.94528515390597482510337621561, −6.24816951954596666481251977932, −5.81788915885788681304204772300, −4.98872981847608146297737901999, −4.38117081296263838315997994326, −3.91980746641886701342722104183, −2.82945256374248667635770934636, −1.65936589640040220021011848136, −0.28550085198741866989837267770,
0.28550085198741866989837267770, 1.65936589640040220021011848136, 2.82945256374248667635770934636, 3.91980746641886701342722104183, 4.38117081296263838315997994326, 4.98872981847608146297737901999, 5.81788915885788681304204772300, 6.24816951954596666481251977932, 6.94528515390597482510337621561, 7.58535490667806779030229944011
