| L(s) = 1 | − 2·2-s + 2·4-s − 5-s − 3·7-s + 2·10-s + 5·11-s + 13-s + 6·14-s − 4·16-s − 5·17-s + 2·19-s − 2·20-s − 10·22-s + 23-s + 25-s − 2·26-s − 6·28-s − 10·29-s − 2·31-s + 8·32-s + 10·34-s + 3·35-s − 3·37-s − 4·38-s + 9·41-s − 4·43-s + 10·44-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.447·5-s − 1.13·7-s + 0.632·10-s + 1.50·11-s + 0.277·13-s + 1.60·14-s − 16-s − 1.21·17-s + 0.458·19-s − 0.447·20-s − 2.13·22-s + 0.208·23-s + 1/5·25-s − 0.392·26-s − 1.13·28-s − 1.85·29-s − 0.359·31-s + 1.41·32-s + 1.71·34-s + 0.507·35-s − 0.493·37-s − 0.648·38-s + 1.40·41-s − 0.609·43-s + 1.50·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−9.975565540863668498254961727135, −9.175134422877607002063538220098, −8.923978077954925138616609003816, −7.66337536315553181125482042712, −6.88984608666809355423599409217, −6.15614460284795576397102652158, −4.40005354130592369090367498749, −3.33489564696074388776632712954, −1.62636499494931476921849119993, 0,
1.62636499494931476921849119993, 3.33489564696074388776632712954, 4.40005354130592369090367498749, 6.15614460284795576397102652158, 6.88984608666809355423599409217, 7.66337536315553181125482042712, 8.923978077954925138616609003816, 9.175134422877607002063538220098, 9.975565540863668498254961727135
