| L(s) = 1 | − 0.209·3-s − 0.418·7-s − 2.95·9-s − 2.65·11-s − 2.81·13-s + 3.81·17-s − 5.26·19-s + 0.0874·21-s + 9.00·23-s + 1.24·27-s + 7.77·29-s − 5.49·31-s + 0.554·33-s + 6.80·37-s + 0.589·39-s + 5.97·41-s − 5.47·43-s + 0.159·47-s − 6.82·49-s − 0.796·51-s + 5.35·53-s + 1.10·57-s + 11.5·59-s + 13.1·61-s + 1.23·63-s + 3.39·67-s − 1.88·69-s + ⋯ |
| L(s) = 1 | − 0.120·3-s − 0.158·7-s − 0.985·9-s − 0.800·11-s − 0.781·13-s + 0.924·17-s − 1.20·19-s + 0.0190·21-s + 1.87·23-s + 0.239·27-s + 1.44·29-s − 0.986·31-s + 0.0965·33-s + 1.11·37-s + 0.0943·39-s + 0.932·41-s − 0.835·43-s + 0.0232·47-s − 0.975·49-s − 0.111·51-s + 0.735·53-s + 0.145·57-s + 1.49·59-s + 1.67·61-s + 0.155·63-s + 0.414·67-s − 0.226·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.209T + 3T^{2} \) |
| 7 | \( 1 + 0.418T + 7T^{2} \) |
| 11 | \( 1 + 2.65T + 11T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 - 3.81T + 17T^{2} \) |
| 19 | \( 1 + 5.26T + 19T^{2} \) |
| 23 | \( 1 - 9.00T + 23T^{2} \) |
| 29 | \( 1 - 7.77T + 29T^{2} \) |
| 31 | \( 1 + 5.49T + 31T^{2} \) |
| 37 | \( 1 - 6.80T + 37T^{2} \) |
| 41 | \( 1 - 5.97T + 41T^{2} \) |
| 43 | \( 1 + 5.47T + 43T^{2} \) |
| 47 | \( 1 - 0.159T + 47T^{2} \) |
| 53 | \( 1 - 5.35T + 53T^{2} \) |
| 59 | \( 1 - 11.5T + 59T^{2} \) |
| 61 | \( 1 - 13.1T + 61T^{2} \) |
| 67 | \( 1 - 3.39T + 67T^{2} \) |
| 71 | \( 1 - 0.116T + 71T^{2} \) |
| 73 | \( 1 + 13.1T + 73T^{2} \) |
| 79 | \( 1 + 5.07T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 0.191T + 89T^{2} \) |
| 97 | \( 1 - 9.56T + 97T^{2} \) |
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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
−7.26392939744588563817564709090, −6.69009095085769211766441381270, −5.86802527245458911590254909225, −5.25395755443087821251713957739, −4.74075226626595515097200567101, −3.73909104352849250297726942067, −2.77132367440770578086231861088, −2.50274199042082219934868581026, −1.06343121186708167212508155866, 0,
1.06343121186708167212508155866, 2.50274199042082219934868581026, 2.77132367440770578086231861088, 3.73909104352849250297726942067, 4.74075226626595515097200567101, 5.25395755443087821251713957739, 5.86802527245458911590254909225, 6.69009095085769211766441381270, 7.26392939744588563817564709090
