| L(s) = 1 | − 3-s − 5-s + 0.585·7-s + 9-s + 2.82·11-s − 2.58·13-s + 15-s − 4·17-s − 2.82·19-s − 0.585·21-s + 6·23-s + 25-s − 27-s + 2.24·29-s − 31-s − 2.82·33-s − 0.585·35-s − 1.41·37-s + 2.58·39-s − 0.828·41-s + 11.3·43-s − 45-s + 4.82·47-s − 6.65·49-s + 4·51-s − 4·53-s − 2.82·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.221·7-s + 0.333·9-s + 0.852·11-s − 0.717·13-s + 0.258·15-s − 0.970·17-s − 0.648·19-s − 0.127·21-s + 1.25·23-s + 0.200·25-s − 0.192·27-s + 0.416·29-s − 0.179·31-s − 0.492·33-s − 0.0990·35-s − 0.232·37-s + 0.414·39-s − 0.129·41-s + 1.72·43-s − 0.149·45-s + 0.704·47-s − 0.950·49-s + 0.560·51-s − 0.549·53-s − 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.269684696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.269684696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 0.585T + 7T^{2} \) |
| 11 | \( 1 - 2.82T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 2.82T + 19T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 - 2.24T + 29T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 41 | \( 1 + 0.828T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 4T + 53T^{2} \) |
| 59 | \( 1 - 0.242T + 59T^{2} \) |
| 61 | \( 1 + 10.4T + 61T^{2} \) |
| 67 | \( 1 + 3.89T + 67T^{2} \) |
| 71 | \( 1 + 9.89T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 - 14.4T + 79T^{2} \) |
| 83 | \( 1 - 0.343T + 83T^{2} \) |
| 89 | \( 1 - 5.07T + 89T^{2} \) |
| 97 | \( 1 - 15.6T + 97T^{2} \) |
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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
−7.67909629624958015939584541888, −7.21490170178832130939205784765, −6.48579704021006542156809930121, −5.94023812137233445137466139151, −4.78770510347286313383082424829, −4.59267606434569748800514349621, −3.66624845190436931284436502811, −2.68694936538832507762125830351, −1.68981471563679820153492271933, −0.59018378522351730703256707559,
0.59018378522351730703256707559, 1.68981471563679820153492271933, 2.68694936538832507762125830351, 3.66624845190436931284436502811, 4.59267606434569748800514349621, 4.78770510347286313383082424829, 5.94023812137233445137466139151, 6.48579704021006542156809930121, 7.21490170178832130939205784765, 7.67909629624958015939584541888
