| L(s) = 1 | + 4-s − 7-s + 9-s + 16-s − 23-s + 25-s − 28-s − 29-s − 31-s + 36-s − 53-s − 63-s + 64-s + 81-s − 83-s − 89-s − 92-s − 97-s + 100-s − 101-s − 112-s − 116-s + ⋯ |
| L(s) = 1 | + 4-s − 7-s + 9-s + 16-s − 23-s + 25-s − 28-s − 29-s − 31-s + 36-s − 53-s − 63-s + 64-s + 81-s − 83-s − 89-s − 92-s − 97-s + 100-s − 101-s − 112-s − 116-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 643 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.100064534\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.100064534\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 643 | \( 1 - T \) |
| good | 2 | \( ( 1 - T )( 1 + T ) \) |
| 3 | \( ( 1 - T )( 1 + T ) \) |
| 5 | \( ( 1 - T )( 1 + T ) \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( ( 1 - T )( 1 + T ) \) |
| 13 | \( ( 1 - T )( 1 + T ) \) |
| 17 | \( ( 1 - T )( 1 + T ) \) |
| 19 | \( ( 1 - T )( 1 + T ) \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 37 | \( ( 1 - T )( 1 + T ) \) |
| 41 | \( ( 1 - T )( 1 + T ) \) |
| 43 | \( ( 1 - T )( 1 + T ) \) |
| 47 | \( ( 1 - T )( 1 + T ) \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( ( 1 - T )( 1 + T ) \) |
| 61 | \( ( 1 - T )( 1 + T ) \) |
| 67 | \( ( 1 - T )( 1 + T ) \) |
| 71 | \( ( 1 - T )( 1 + T ) \) |
| 73 | \( ( 1 - T )( 1 + T ) \) |
| 79 | \( ( 1 - T )( 1 + T ) \) |
| 83 | \( 1 + T + T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−10.71263538745193220421839502131, −9.992562171607717376658605155789, −9.256767226866823683164663452628, −7.949098978020109802468250783181, −7.09811732045868397920703922587, −6.48973698624041496127341694059, −5.51593260392634103602563015577, −4.05592133023485231684242479158, −3.04761064021838015210864751315, −1.73887166628522477691175598862,
1.73887166628522477691175598862, 3.04761064021838015210864751315, 4.05592133023485231684242479158, 5.51593260392634103602563015577, 6.48973698624041496127341694059, 7.09811732045868397920703922587, 7.949098978020109802468250783181, 9.256767226866823683164663452628, 9.992562171607717376658605155789, 10.71263538745193220421839502131
