| L(s) = 1 | − 20.0·3-s − 46.6·5-s + 25.6·7-s + 157.·9-s + 355.·11-s − 865.·13-s + 933.·15-s − 1.62e3·17-s − 1.54e3·19-s − 513.·21-s + 4.74e3·23-s − 948.·25-s + 1.71e3·27-s + 2.02e3·29-s − 907.·31-s − 7.10e3·33-s − 1.19e3·35-s − 1.20e4·37-s + 1.73e4·39-s − 5.58e3·41-s − 3.32e3·43-s − 7.34e3·45-s + 1.07e4·47-s − 1.61e4·49-s + 3.24e4·51-s + 1.79e4·53-s − 1.65e4·55-s + ⋯ |
| L(s) = 1 | − 1.28·3-s − 0.834·5-s + 0.197·7-s + 0.648·9-s + 0.884·11-s − 1.42·13-s + 1.07·15-s − 1.36·17-s − 0.982·19-s − 0.253·21-s + 1.87·23-s − 0.303·25-s + 0.451·27-s + 0.446·29-s − 0.169·31-s − 1.13·33-s − 0.165·35-s − 1.44·37-s + 1.82·39-s − 0.518·41-s − 0.274·43-s − 0.540·45-s + 0.712·47-s − 0.960·49-s + 1.74·51-s + 0.876·53-s − 0.738·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.2039075163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2039075163\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 257 | \( 1 + 6.60e4T \) |
| good | 3 | \( 1 + 20.0T + 243T^{2} \) |
| 5 | \( 1 + 46.6T + 3.12e3T^{2} \) |
| 7 | \( 1 - 25.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 355.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 865.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.62e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.54e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.74e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.02e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 907.T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.20e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.58e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.32e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.07e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.79e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.40e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.38e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.26e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.77e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.16e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.53e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.22e5T + 8.58e9T^{2} \) |
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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
−9.128755407750368431998893223034, −8.469168078862306390735841703273, −7.09264441529439142198302476461, −6.89160462204022505007744120042, −5.77483536681370398528964032813, −4.73067591270793100268750466961, −4.34882950223365877172865656048, −2.92731194443857764945637006117, −1.55222076783713472120670691235, −0.21159113733277131243588691925,
0.21159113733277131243588691925, 1.55222076783713472120670691235, 2.92731194443857764945637006117, 4.34882950223365877172865656048, 4.73067591270793100268750466961, 5.77483536681370398528964032813, 6.89160462204022505007744120042, 7.09264441529439142198302476461, 8.469168078862306390735841703273, 9.128755407750368431998893223034
