| L(s) = 1 | + 2.82·3-s + 5-s − 0.0498·7-s + 4.98·9-s − 4.45·11-s − 2.43·13-s + 2.82·15-s − 2.48·17-s + 5.16·19-s − 0.140·21-s + 4.99·23-s + 25-s + 5.62·27-s + 2.59·29-s + 7.31·31-s − 12.5·33-s − 0.0498·35-s + 5.13·37-s − 6.87·39-s + 9.11·41-s + 0.543·43-s + 4.98·45-s + 9.63·47-s − 6.99·49-s − 7.01·51-s − 10.7·53-s − 4.45·55-s + ⋯ |
| L(s) = 1 | + 1.63·3-s + 0.447·5-s − 0.0188·7-s + 1.66·9-s − 1.34·11-s − 0.674·13-s + 0.729·15-s − 0.601·17-s + 1.18·19-s − 0.0307·21-s + 1.04·23-s + 0.200·25-s + 1.08·27-s + 0.481·29-s + 1.31·31-s − 2.19·33-s − 0.00842·35-s + 0.844·37-s − 1.10·39-s + 1.42·41-s + 0.0828·43-s + 0.743·45-s + 1.40·47-s − 0.999·49-s − 0.981·51-s − 1.47·53-s − 0.600·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.049203202\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.049203202\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 89 | \( 1 - T \) |
| good | 3 | \( 1 - 2.82T + 3T^{2} \) |
| 7 | \( 1 + 0.0498T + 7T^{2} \) |
| 11 | \( 1 + 4.45T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 + 2.48T + 17T^{2} \) |
| 19 | \( 1 - 5.16T + 19T^{2} \) |
| 23 | \( 1 - 4.99T + 23T^{2} \) |
| 29 | \( 1 - 2.59T + 29T^{2} \) |
| 31 | \( 1 - 7.31T + 31T^{2} \) |
| 37 | \( 1 - 5.13T + 37T^{2} \) |
| 41 | \( 1 - 9.11T + 41T^{2} \) |
| 43 | \( 1 - 0.543T + 43T^{2} \) |
| 47 | \( 1 - 9.63T + 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 - 12.6T + 59T^{2} \) |
| 61 | \( 1 + 2.47T + 61T^{2} \) |
| 67 | \( 1 - 5.36T + 67T^{2} \) |
| 71 | \( 1 + 9.58T + 71T^{2} \) |
| 73 | \( 1 - 2.86T + 73T^{2} \) |
| 79 | \( 1 + 6.62T + 79T^{2} \) |
| 83 | \( 1 - 12.0T + 83T^{2} \) |
| 97 | \( 1 - 4.82T + 97T^{2} \) |
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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
−7.947172833270048500612460605445, −7.47341290670049593416832651631, −6.79712191919945321087427283581, −5.78133385751310335452316964031, −4.93403285364075839161296005686, −4.35486973577311909261561224487, −3.18242261680467141192771749380, −2.72691034326708501281386355109, −2.19957724867441544390766155122, −0.951493880314348108847753355724,
0.951493880314348108847753355724, 2.19957724867441544390766155122, 2.72691034326708501281386355109, 3.18242261680467141192771749380, 4.35486973577311909261561224487, 4.93403285364075839161296005686, 5.78133385751310335452316964031, 6.79712191919945321087427283581, 7.47341290670049593416832651631, 7.947172833270048500612460605445
