L(s) = 1 | + 3-s − 5-s − 7-s + 9-s + 4·11-s − 13-s − 15-s − 2·17-s − 21-s + 4·23-s + 25-s + 27-s − 10·29-s − 4·31-s + 4·33-s + 35-s + 6·37-s − 39-s − 2·41-s − 4·43-s − 45-s + 49-s − 2·51-s − 14·53-s − 4·55-s + 4·59-s − 10·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.277·13-s − 0.258·15-s − 0.485·17-s − 0.218·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s + 0.696·33-s + 0.169·35-s + 0.986·37-s − 0.160·39-s − 0.312·41-s − 0.609·43-s − 0.149·45-s + 1/7·49-s − 0.280·51-s − 1.92·53-s − 0.539·55-s + 0.520·59-s − 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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
−16.68944265368387, −16.35508338273716, −15.48042483682562, −15.02479534152539, −14.64704142402422, −14.02505589141623, −13.30296958840780, −12.84041492861318, −12.30773425260278, −11.43438653362622, −11.21526568438913, −10.36104240064980, −9.511565158281282, −9.203746240448025, −8.700267821057358, −7.769273980007437, −7.360822792245006, −6.632515146106294, −6.079053729775924, −5.080639174135736, −4.366500371161671, −3.659866362112842, −3.158784229839195, −2.130621580473238, −1.302642364193455, 0,
1.302642364193455, 2.130621580473238, 3.158784229839195, 3.659866362112842, 4.366500371161671, 5.080639174135736, 6.079053729775924, 6.632515146106294, 7.360822792245006, 7.769273980007437, 8.700267821057358, 9.203746240448025, 9.511565158281282, 10.36104240064980, 11.21526568438913, 11.43438653362622, 12.30773425260278, 12.84041492861318, 13.30296958840780, 14.02505589141623, 14.64704142402422, 15.02479534152539, 15.48042483682562, 16.35508338273716, 16.68944265368387