L(s) = 1 | − 7-s − 6·11-s − 2·13-s − 2·17-s − 4·19-s + 4·23-s + 2·29-s + 2·31-s + 10·37-s + 6·41-s − 2·43-s − 2·47-s + 49-s + 6·53-s + 4·59-s − 12·61-s + 10·67-s − 12·71-s + 2·73-s + 6·77-s + 16·79-s + 12·83-s + 14·89-s + 2·91-s − 18·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 1.80·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 0.371·29-s + 0.359·31-s + 1.64·37-s + 0.937·41-s − 0.304·43-s − 0.291·47-s + 1/7·49-s + 0.824·53-s + 0.520·59-s − 1.53·61-s + 1.22·67-s − 1.42·71-s + 0.234·73-s + 0.683·77-s + 1.80·79-s + 1.31·83-s + 1.48·89-s + 0.209·91-s − 1.82·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−15.51348087489482, −15.10188882441420, −14.78676592501211, −13.89001842926704, −13.35268484084209, −12.99394791751469, −12.53933260778492, −11.94912315666613, −11.04040167696378, −10.85347514621506, −10.16885659658088, −9.714392708232562, −9.014176085808490, −8.444329752999385, −7.748743518379127, −7.449115064816102, −6.556801143327780, −6.134502540202862, −5.302143615660665, −4.816223595792053, −4.225961117983937, −3.270406993392790, −2.544229687413557, −2.246668442901821, −0.8730980542688705, 0,
0.8730980542688705, 2.246668442901821, 2.544229687413557, 3.270406993392790, 4.225961117983937, 4.816223595792053, 5.302143615660665, 6.134502540202862, 6.556801143327780, 7.449115064816102, 7.748743518379127, 8.444329752999385, 9.014176085808490, 9.714392708232562, 10.16885659658088, 10.85347514621506, 11.04040167696378, 11.94912315666613, 12.53933260778492, 12.99394791751469, 13.35268484084209, 13.89001842926704, 14.78676592501211, 15.10188882441420, 15.51348087489482