| L(s) = 1 | + 6·9-s + 4·17-s + 6·25-s − 20·41-s − 14·49-s + 12·73-s + 27·81-s − 20·89-s + 36·97-s − 28·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 24·153-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 2·9-s + 0.970·17-s + 6/5·25-s − 3.12·41-s − 2·49-s + 1.40·73-s + 3·81-s − 2.11·89-s + 3.65·97-s − 2.63·113-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 1.94·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.718796454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.718796454\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.24465143591019829183704169599, −11.95867331713989356615905972730, −11.26093909185056800113423925172, −10.76318786599095307120558263935, −10.13053160696446419990753279668, −10.06255711279168335279332587926, −9.511503879403609031118229552499, −8.966760340515443645903341695812, −8.138667714368049024606349572239, −8.033182871671553594656430544981, −7.04783751610773635462797907079, −6.99447575780113373208701869739, −6.41743200731889103929487159669, −5.58010149012749333429319409235, −4.75086858462215078956990844947, −4.72083209570865226203480983593, −3.60827722963180271858468189137, −3.27909722254998121465231940274, −1.97376455261820330387405606640, −1.23246026084251524188760195146,
1.23246026084251524188760195146, 1.97376455261820330387405606640, 3.27909722254998121465231940274, 3.60827722963180271858468189137, 4.72083209570865226203480983593, 4.75086858462215078956990844947, 5.58010149012749333429319409235, 6.41743200731889103929487159669, 6.99447575780113373208701869739, 7.04783751610773635462797907079, 8.033182871671553594656430544981, 8.138667714368049024606349572239, 8.966760340515443645903341695812, 9.511503879403609031118229552499, 10.06255711279168335279332587926, 10.13053160696446419990753279668, 10.76318786599095307120558263935, 11.26093909185056800113423925172, 11.95867331713989356615905972730, 12.24465143591019829183704169599
