| L(s) = 1 | − 0.381·3-s + 5-s − 7-s − 2.85·9-s − 5.09·11-s − 2.23·13-s − 0.381·15-s − 5.38·17-s − 19-s + 0.381·21-s − 4.70·23-s − 4·25-s + 2.23·27-s + 5.85·29-s + 4.61·31-s + 1.94·33-s − 35-s − 6.70·37-s + 0.854·39-s − 11.0·41-s − 0.472·43-s − 2.85·45-s + 8.70·47-s + 49-s + 2.05·51-s − 1.32·53-s − 5.09·55-s + ⋯ |
| L(s) = 1 | − 0.220·3-s + 0.447·5-s − 0.377·7-s − 0.951·9-s − 1.53·11-s − 0.620·13-s − 0.0986·15-s − 1.30·17-s − 0.229·19-s + 0.0833·21-s − 0.981·23-s − 0.800·25-s + 0.430·27-s + 1.08·29-s + 0.829·31-s + 0.338·33-s − 0.169·35-s − 1.10·37-s + 0.136·39-s − 1.73·41-s − 0.0720·43-s − 0.425·45-s + 1.27·47-s + 0.142·49-s + 0.287·51-s − 0.182·53-s − 0.686·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5411487308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5411487308\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 + 5.09T + 11T^{2} \) |
| 13 | \( 1 + 2.23T + 13T^{2} \) |
| 17 | \( 1 + 5.38T + 17T^{2} \) |
| 23 | \( 1 + 4.70T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 - 4.61T + 31T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 0.472T + 43T^{2} \) |
| 47 | \( 1 - 8.70T + 47T^{2} \) |
| 53 | \( 1 + 1.32T + 53T^{2} \) |
| 59 | \( 1 - 2.70T + 59T^{2} \) |
| 61 | \( 1 - 3.47T + 61T^{2} \) |
| 67 | \( 1 - 9.09T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 12.3T + 83T^{2} \) |
| 89 | \( 1 - 10.1T + 89T^{2} \) |
| 97 | \( 1 + 4.70T + 97T^{2} \) |
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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
−7.84603463354390528991459396507, −7.01325854932454351956224155297, −6.32797983846667977445671949667, −5.72798736689698697143935543489, −5.07822728077104275182769266687, −4.43814594818679021550988964113, −3.31900225799846868766661896904, −2.52836848408661493356716141149, −2.04548452184538125360581546211, −0.33264512993748612238798931157,
0.33264512993748612238798931157, 2.04548452184538125360581546211, 2.52836848408661493356716141149, 3.31900225799846868766661896904, 4.43814594818679021550988964113, 5.07822728077104275182769266687, 5.72798736689698697143935543489, 6.32797983846667977445671949667, 7.01325854932454351956224155297, 7.84603463354390528991459396507
