| L(s) = 1 | − 0.323·3-s − 10.9·5-s − 6.41·7-s − 26.8·9-s − 54.0·11-s − 13·13-s + 3.53·15-s − 26.1·17-s − 59.7·19-s + 2.07·21-s + 107.·23-s − 5.65·25-s + 17.4·27-s − 135.·29-s + 83.1·31-s + 17.4·33-s + 70.0·35-s + 98.8·37-s + 4.20·39-s + 436.·41-s − 101.·43-s + 293.·45-s − 94.0·47-s − 301.·49-s + 8.47·51-s + 226.·53-s + 590.·55-s + ⋯ |
| L(s) = 1 | − 0.0622·3-s − 0.977·5-s − 0.346·7-s − 0.996·9-s − 1.48·11-s − 0.277·13-s + 0.0608·15-s − 0.373·17-s − 0.721·19-s + 0.0215·21-s + 0.972·23-s − 0.0452·25-s + 0.124·27-s − 0.865·29-s + 0.481·31-s + 0.0921·33-s + 0.338·35-s + 0.439·37-s + 0.0172·39-s + 1.66·41-s − 0.359·43-s + 0.973·45-s − 0.291·47-s − 0.879·49-s + 0.0232·51-s + 0.587·53-s + 1.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.5243990135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5243990135\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + 13T \) |
| good | 3 | \( 1 + 0.323T + 27T^{2} \) |
| 5 | \( 1 + 10.9T + 125T^{2} \) |
| 7 | \( 1 + 6.41T + 343T^{2} \) |
| 11 | \( 1 + 54.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 26.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 59.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 107.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 135.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 83.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 98.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 436.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 94.0T + 1.03e5T^{2} \) |
| 53 | \( 1 - 226.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 811.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 291.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 551.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 838.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 762.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 252.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.09e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
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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
−9.833498270325033024829005194722, −8.855990637637834936721119343712, −8.046939319752293632134437290816, −7.46210078305654069017768482321, −6.32507422365881582548373358959, −5.36409443900457599284469607223, −4.44029697829553050538199169511, −3.25411189032019217647311284834, −2.41524816107859034315574960911, −0.37539759691737470518153225657,
0.37539759691737470518153225657, 2.41524816107859034315574960911, 3.25411189032019217647311284834, 4.44029697829553050538199169511, 5.36409443900457599284469607223, 6.32507422365881582548373358959, 7.46210078305654069017768482321, 8.046939319752293632134437290816, 8.855990637637834936721119343712, 9.833498270325033024829005194722
