| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s − 11-s − 12-s + 13-s + 14-s + 15-s + 16-s − 17-s + 18-s − 19-s − 20-s − 21-s − 22-s + 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 367 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.614155034\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.614155034\) |
| \(L(1)\) |
\(\approx\) |
\(1.475908214\) |
| \(L(1)\) |
\(\approx\) |
\(1.475908214\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 367 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−24.07295344427227916874467011058, −23.39375800551083146116290828474, −23.04614205162520113168926827583, −21.91466984398112239197876735766, −21.05500027791353624261487685782, −20.46830595644090939713749872424, −19.1496905173913490427105806983, −18.27755729397218562027159172359, −17.18096190396635690138152398530, −16.219261881409954433527728838799, −15.43372193075151330707375233747, −14.89494636863222701186534308176, −13.36101839846742971101565702138, −12.78819201979949167757426663937, −11.61959262889186169649375956399, −11.11011596746567091634282514029, −10.554050395696314949375444037175, −8.4843781011859145068327343938, −7.53450453793685223985425491642, −6.60156249957092905297031518942, −5.48162804846609629609268813022, −4.610836189071464547860425879469, −3.929459366410916690305532924, −2.32486240416487945515243663933, −0.862638631695890076901632988102,
0.862638631695890076901632988102, 2.32486240416487945515243663933, 3.929459366410916690305532924, 4.610836189071464547860425879469, 5.48162804846609629609268813022, 6.60156249957092905297031518942, 7.53450453793685223985425491642, 8.4843781011859145068327343938, 10.554050395696314949375444037175, 11.11011596746567091634282514029, 11.61959262889186169649375956399, 12.78819201979949167757426663937, 13.36101839846742971101565702138, 14.89494636863222701186534308176, 15.43372193075151330707375233747, 16.219261881409954433527728838799, 17.18096190396635690138152398530, 18.27755729397218562027159172359, 19.1496905173913490427105806983, 20.46830595644090939713749872424, 21.05500027791353624261487685782, 21.91466984398112239197876735766, 23.04614205162520113168926827583, 23.39375800551083146116290828474, 24.07295344427227916874467011058
