L(s) = 1 | − 1.73·3-s + 5.19·7-s + 2.99·9-s + 7i·13-s − 5.19i·19-s − 9·21-s − 5.19·27-s − 1.73i·31-s + 10i·37-s − 12.1i·39-s + 1.73·43-s + 20·49-s + 9i·57-s − 61-s + 15.5·63-s + ⋯ |
L(s) = 1 | − 1.00·3-s + 1.96·7-s + 0.999·9-s + 1.94i·13-s − 1.19i·19-s − 1.96·21-s − 1.00·27-s − 0.311i·31-s + 1.64i·37-s − 1.94i·39-s + 0.264·43-s + 2.85·49-s + 1.19i·57-s − 0.128·61-s + 1.96·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.483583614\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483583614\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - 5.19T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 7iT - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 1.73iT - 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 1.73T + 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + T + 61T^{2} \) |
| 67 | \( 1 - 12.1T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 17.3iT - 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 19iT - 97T^{2} \) |
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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
−9.904006349762734557296986782395, −8.990878629567426067556868903106, −8.173833551262057705127484575778, −7.18761346457029572424150468654, −6.59592566059332526814211546733, −5.41644707696064774491957438730, −4.67522734590156957256078108317, −4.20030273483510850580687894976, −2.16872630018437580509840248141, −1.23862990188887445243971717259,
0.883859279042136434844977447775, 2.01688487166431710759503963508, 3.69742887542673503918349864196, 4.77922210016807609782435976090, 5.40719083602236903772530540579, 6.02219620553493737625539385078, 7.48578905327142237613297480225, 7.81290632483325185812728639685, 8.686561821937392280732477247688, 9.984032236899566540939804928529