| L(s) = 1 | + 3-s − 2·4-s − 7-s + 9-s − 2·12-s + 4·16-s − 6·17-s − 19-s − 21-s + 6·23-s − 5·25-s + 27-s + 2·28-s + 7·31-s − 2·36-s − 5·37-s + 4·43-s + 6·47-s + 4·48-s − 6·49-s − 6·51-s − 57-s + 6·59-s + 61-s − 63-s − 8·64-s − 5·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s + 1/3·9-s − 0.577·12-s + 16-s − 1.45·17-s − 0.229·19-s − 0.218·21-s + 1.25·23-s − 25-s + 0.192·27-s + 0.377·28-s + 1.25·31-s − 1/3·36-s − 0.821·37-s + 0.609·43-s + 0.875·47-s + 0.577·48-s − 6/7·49-s − 0.840·51-s − 0.132·57-s + 0.781·59-s + 0.128·61-s − 0.125·63-s − 64-s − 0.610·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61347 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 7 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 18 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
| 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
−14.46997283353544, −13.85953750252588, −13.58619586865843, −13.14555254600258, −12.70649151079161, −12.16005655008649, −11.53127216290532, −10.85597762701478, −10.41474722301332, −9.780079197255395, −9.337024702869771, −8.923532151946102, −8.444318936541381, −7.976422238763618, −7.286948181865011, −6.695625589657726, −6.206366359102985, −5.391247077346834, −4.898005969591190, −4.199821564264871, −3.918336427996354, −3.074995033757832, −2.559996399987458, −1.729854172585306, −0.8436685276696063, 0,
0.8436685276696063, 1.729854172585306, 2.559996399987458, 3.074995033757832, 3.918336427996354, 4.199821564264871, 4.898005969591190, 5.391247077346834, 6.206366359102985, 6.695625589657726, 7.286948181865011, 7.976422238763618, 8.444318936541381, 8.923532151946102, 9.337024702869771, 9.780079197255395, 10.41474722301332, 10.85597762701478, 11.53127216290532, 12.16005655008649, 12.70649151079161, 13.14555254600258, 13.58619586865843, 13.85953750252588, 14.46997283353544
