| L(s) = 1 | + 0.163·3-s − 5.36·5-s + 7·7-s − 26.9·9-s − 11·11-s − 42.9·13-s − 0.877·15-s − 60.8·17-s − 140.·19-s + 1.14·21-s + 91.3·23-s − 96.1·25-s − 8.81·27-s + 260.·29-s + 259.·31-s − 1.79·33-s − 37.5·35-s + 359.·37-s − 7.01·39-s − 320.·41-s + 92.3·43-s + 144.·45-s − 67.4·47-s + 49·49-s − 9.94·51-s − 246.·53-s + 59.0·55-s + ⋯ |
| L(s) = 1 | + 0.0314·3-s − 0.480·5-s + 0.377·7-s − 0.999·9-s − 0.301·11-s − 0.916·13-s − 0.0150·15-s − 0.868·17-s − 1.69·19-s + 0.0118·21-s + 0.828·23-s − 0.769·25-s − 0.0628·27-s + 1.67·29-s + 1.50·31-s − 0.00948·33-s − 0.181·35-s + 1.59·37-s − 0.0288·39-s − 1.21·41-s + 0.327·43-s + 0.479·45-s − 0.209·47-s + 0.142·49-s − 0.0273·51-s − 0.637·53-s + 0.144·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.117539877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.117539877\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 + 11T \) |
| good | 3 | \( 1 - 0.163T + 27T^{2} \) |
| 5 | \( 1 + 5.36T + 125T^{2} \) |
| 13 | \( 1 + 42.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 60.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 91.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 260.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 259.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 359.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 320.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 92.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 67.4T + 1.03e5T^{2} \) |
| 53 | \( 1 + 246.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 475.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 799.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 725.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 544.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 580.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 402.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.99e2T + 9.12e5T^{2} \) |
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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
−9.221085721195111039374415378102, −8.344406528040587115823427035801, −7.989177844510453463024121554777, −6.79049005049065676486821112400, −6.11325098309941533167303827453, −4.87111201137943504660084390387, −4.36862681366438678134819733004, −2.95668971419341524536170272093, −2.21277724232716814138879751755, −0.51053792493050557233320647549,
0.51053792493050557233320647549, 2.21277724232716814138879751755, 2.95668971419341524536170272093, 4.36862681366438678134819733004, 4.87111201137943504660084390387, 6.11325098309941533167303827453, 6.79049005049065676486821112400, 7.989177844510453463024121554777, 8.344406528040587115823427035801, 9.221085721195111039374415378102
