| L(s) = 1 | − 3.33·3-s − 0.131·5-s − 7-s + 8.11·9-s + 11-s − 13-s + 0.437·15-s + 4.35·17-s + 0.765·19-s + 3.33·21-s − 2.35·23-s − 4.98·25-s − 17.0·27-s + 1.55·29-s + 3.44·31-s − 3.33·33-s + 0.131·35-s − 6.91·37-s + 3.33·39-s + 7.35·41-s − 9.25·43-s − 1.06·45-s − 2.35·47-s + 49-s − 14.5·51-s + 6.98·53-s − 0.131·55-s + ⋯ |
| L(s) = 1 | − 1.92·3-s − 0.0587·5-s − 0.377·7-s + 2.70·9-s + 0.301·11-s − 0.277·13-s + 0.113·15-s + 1.05·17-s + 0.175·19-s + 0.727·21-s − 0.490·23-s − 0.996·25-s − 3.28·27-s + 0.287·29-s + 0.617·31-s − 0.580·33-s + 0.0222·35-s − 1.13·37-s + 0.533·39-s + 1.14·41-s − 1.41·43-s − 0.158·45-s − 0.343·47-s + 0.142·49-s − 2.03·51-s + 0.959·53-s − 0.0177·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 3.33T + 3T^{2} \) |
| 5 | \( 1 + 0.131T + 5T^{2} \) |
| 17 | \( 1 - 4.35T + 17T^{2} \) |
| 19 | \( 1 - 0.765T + 19T^{2} \) |
| 23 | \( 1 + 2.35T + 23T^{2} \) |
| 29 | \( 1 - 1.55T + 29T^{2} \) |
| 31 | \( 1 - 3.44T + 31T^{2} \) |
| 37 | \( 1 + 6.91T + 37T^{2} \) |
| 41 | \( 1 - 7.35T + 41T^{2} \) |
| 43 | \( 1 + 9.25T + 43T^{2} \) |
| 47 | \( 1 + 2.35T + 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 + 0.151T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 + 7.77T + 67T^{2} \) |
| 71 | \( 1 + 0.493T + 71T^{2} \) |
| 73 | \( 1 + 3.05T + 73T^{2} \) |
| 79 | \( 1 - 1.38T + 79T^{2} \) |
| 83 | \( 1 - 4.49T + 83T^{2} \) |
| 89 | \( 1 - 6.66T + 89T^{2} \) |
| 97 | \( 1 - 18.6T + 97T^{2} \) |
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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
−7.37197809354705853731954682665, −6.55752207611660099222820121883, −6.10180409252579901639221772528, −5.48285927569792879757029122487, −4.85359003607718243145294888320, −4.10978772463979062468812194410, −3.32053374108001886652616396946, −1.90466366762016866848537460151, −0.982738684798633869226936440534, 0,
0.982738684798633869226936440534, 1.90466366762016866848537460151, 3.32053374108001886652616396946, 4.10978772463979062468812194410, 4.85359003607718243145294888320, 5.48285927569792879757029122487, 6.10180409252579901639221772528, 6.55752207611660099222820121883, 7.37197809354705853731954682665
