| L(s) = 1 | − 2.68·3-s + 0.526·5-s + 7-s + 4.21·9-s − 0.224·11-s − 2.09·13-s − 1.41·15-s − 3.79·17-s + 4.16·19-s − 2.68·21-s + 1.95·23-s − 4.72·25-s − 3.25·27-s − 0.0431·29-s + 9.16·31-s + 0.603·33-s + 0.526·35-s − 8.18·37-s + 5.63·39-s + 10.9·41-s − 3.71·43-s + 2.21·45-s − 7.79·47-s + 49-s + 10.1·51-s − 3.12·53-s − 0.118·55-s + ⋯ |
| L(s) = 1 | − 1.55·3-s + 0.235·5-s + 0.377·7-s + 1.40·9-s − 0.0677·11-s − 0.582·13-s − 0.365·15-s − 0.921·17-s + 0.955·19-s − 0.586·21-s + 0.408·23-s − 0.944·25-s − 0.626·27-s − 0.00800·29-s + 1.64·31-s + 0.105·33-s + 0.0890·35-s − 1.34·37-s + 0.902·39-s + 1.70·41-s − 0.566·43-s + 0.330·45-s − 1.13·47-s + 0.142·49-s + 1.42·51-s − 0.429·53-s − 0.0159·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 3 | \( 1 + 2.68T + 3T^{2} \) |
| 5 | \( 1 - 0.526T + 5T^{2} \) |
| 11 | \( 1 + 0.224T + 11T^{2} \) |
| 13 | \( 1 + 2.09T + 13T^{2} \) |
| 17 | \( 1 + 3.79T + 17T^{2} \) |
| 19 | \( 1 - 4.16T + 19T^{2} \) |
| 23 | \( 1 - 1.95T + 23T^{2} \) |
| 29 | \( 1 + 0.0431T + 29T^{2} \) |
| 31 | \( 1 - 9.16T + 31T^{2} \) |
| 37 | \( 1 + 8.18T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 3.71T + 43T^{2} \) |
| 47 | \( 1 + 7.79T + 47T^{2} \) |
| 53 | \( 1 + 3.12T + 53T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + 7.03T + 61T^{2} \) |
| 67 | \( 1 - 6.25T + 67T^{2} \) |
| 71 | \( 1 - 14.5T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 0.175T + 83T^{2} \) |
| 89 | \( 1 - 6.79T + 89T^{2} \) |
| 97 | \( 1 - 8.80T + 97T^{2} \) |
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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
−7.47594910357051970333664181618, −6.63311646319492598759887717794, −6.23125384848268329256149833177, −5.39477979565049650808961907018, −4.88964643653430923164034582441, −4.33487374261743683270071915073, −3.15190735123374802579376958062, −2.07071933726981455115712874263, −1.07713227106209609363843258182, 0,
1.07713227106209609363843258182, 2.07071933726981455115712874263, 3.15190735123374802579376958062, 4.33487374261743683270071915073, 4.88964643653430923164034582441, 5.39477979565049650808961907018, 6.23125384848268329256149833177, 6.63311646319492598759887717794, 7.47594910357051970333664181618
