| L(s) = 1 | − 2-s − 5-s + 2·7-s + 8-s + 9-s + 10-s − 11-s − 2·14-s − 16-s − 18-s − 19-s + 22-s + 2·23-s − 29-s − 2·35-s + 2·37-s + 38-s − 40-s − 41-s − 45-s − 2·46-s + 3·49-s + 55-s + 2·56-s + 58-s + 2·59-s + 2·61-s + ⋯ |
| L(s) = 1 | − 2-s − 5-s + 2·7-s + 8-s + 9-s + 10-s − 11-s − 2·14-s − 16-s − 18-s − 19-s + 22-s + 2·23-s − 29-s − 2·35-s + 2·37-s + 38-s − 40-s − 41-s − 45-s − 2·46-s + 3·49-s + 55-s + 2·56-s + 58-s + 2·59-s + 2·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6437872217\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6437872217\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1399 | \( 1 - T \) |
| good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( 1 + T + T^{2} \) |
| 7 | \( ( 1 - T )^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( 1 + T + T^{2} \) |
| 23 | \( ( 1 - T )^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( ( 1 - T )( 1 + T ) \) |
| 37 | \( ( 1 - T )^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( ( 1 - T )( 1 + T ) \) |
| 59 | \( ( 1 - T )^{2} \) |
| 61 | \( ( 1 - T )^{2} \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( ( 1 - T )( 1 + T ) \) |
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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
−9.759731041178988968598799558154, −8.568056651444912567499038923847, −8.344650559021080866519740197442, −7.44921861284023614347859999688, −7.20374333995643615729740770239, −5.32340058760466289510035506056, −4.62579794010255100777731712712, −4.03683493161433269498242535941, −2.21365758376716401334781986928, −1.08660344305958077950471814441,
1.08660344305958077950471814441, 2.21365758376716401334781986928, 4.03683493161433269498242535941, 4.62579794010255100777731712712, 5.32340058760466289510035506056, 7.20374333995643615729740770239, 7.44921861284023614347859999688, 8.344650559021080866519740197442, 8.568056651444912567499038923847, 9.759731041178988968598799558154
