L(s) = 1 | − 2·2-s + 3·4-s − 7-s − 4·8-s − 5·9-s + 2·14-s + 5·16-s + 10·18-s − 12·23-s − 25-s − 3·28-s − 6·32-s − 15·36-s + 4·37-s + 16·43-s + 24·46-s − 6·49-s + 2·50-s + 12·53-s + 4·56-s + 5·63-s + 7·64-s − 8·67-s − 18·71-s + 20·72-s − 8·74-s + 2·79-s + ⋯ |
L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.377·7-s − 1.41·8-s − 5/3·9-s + 0.534·14-s + 5/4·16-s + 2.35·18-s − 2.50·23-s − 1/5·25-s − 0.566·28-s − 1.06·32-s − 5/2·36-s + 0.657·37-s + 2.43·43-s + 3.53·46-s − 6/7·49-s + 0.282·50-s + 1.64·53-s + 0.534·56-s + 0.629·63-s + 7/8·64-s − 0.977·67-s − 2.13·71-s + 2.35·72-s − 0.929·74-s + 0.225·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1223236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1223236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 79 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
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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
−8.008923035003699396981349994067, −7.48344544305932962254865306275, −7.15971126364765784239855073288, −6.43698619599801084891178808154, −6.08907419502922611955487703530, −5.66674862285945396423871921055, −5.64461215490222988307046707304, −4.50300100754317813912593159477, −4.11354389527123202675815643778, −3.35403695021279589879198759951, −2.91410117506949580909848279609, −2.30932045676551205221388953943, −1.90147284566448759002651582081, −0.77056399968604496595837882445, 0,
0.77056399968604496595837882445, 1.90147284566448759002651582081, 2.30932045676551205221388953943, 2.91410117506949580909848279609, 3.35403695021279589879198759951, 4.11354389527123202675815643778, 4.50300100754317813912593159477, 5.64461215490222988307046707304, 5.66674862285945396423871921055, 6.08907419502922611955487703530, 6.43698619599801084891178808154, 7.15971126364765784239855073288, 7.48344544305932962254865306275, 8.008923035003699396981349994067