| L(s) = 1 | − 3.46·5-s + (−1 + 2.44i)7-s + 4.24i·11-s + 4.89i·13-s − 3.46·17-s − 4.89i·19-s − 4.24i·23-s + 6.99·25-s + 4.24i·29-s + (3.46 − 8.48i)35-s − 8·37-s + 3.46·41-s − 2·43-s + 6.92·47-s + (−4.99 − 4.89i)49-s + ⋯ |
| L(s) = 1 | − 1.54·5-s + (−0.377 + 0.925i)7-s + 1.27i·11-s + 1.35i·13-s − 0.840·17-s − 1.12i·19-s − 0.884i·23-s + 1.39·25-s + 0.787i·29-s + (0.585 − 1.43i)35-s − 1.31·37-s + 0.541·41-s − 0.304·43-s + 1.01·47-s + (−0.714 − 0.699i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.287052 + 0.523529i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.287052 + 0.523529i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (1 - 2.44i)T \) |
| good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 4.89iT - 19T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 - 4.24iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + 8T + 37T^{2} \) |
| 41 | \( 1 - 3.46T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 12.7iT - 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 9.79iT - 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 4.24iT - 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 + 6.92T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 4.89iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.14644714919715094421580642785, −11.67861299369181063982752187623, −10.63769351616237099499138518733, −9.193176370201829518409544224170, −8.668363417051754932643940315032, −7.27696966240240158037378687916, −6.68390668405292913129683867863, −4.87004105115356230309308212383, −4.04238038388355147148163060953, −2.43217406373457657783319091889,
0.46606084626972137445853824375, 3.34581103770000706355534884433, 3.96386430145868860875921572462, 5.54183921971613717507141750352, 6.90068257714039438901093338788, 7.911000192099818363452667037952, 8.444759433140861181874874284930, 10.00204980408548105012487791037, 10.92327773858664145596923125900, 11.56586037001438557656639339046
