L(s) = 1 | − 3·9-s − 16·29-s + 6·41-s + 10·49-s − 20·61-s − 18·89-s + 12·101-s − 28·109-s − 21·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
L(s) = 1 | − 9-s − 2.97·29-s + 0.937·41-s + 10/7·49-s − 2.56·61-s − 1.90·89-s + 1.19·101-s − 2.68·109-s − 1.90·121-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 + 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 + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−8.955462753706152130249428036919, −8.701941492535567968840109922622, −7.916186872483072797338932024160, −7.60840457104033872823456309121, −7.18964714691018416464928980947, −6.47166977500015585273964086632, −5.87134480744638375141626815412, −5.61017205144009625913946893858, −5.06850257247920738912499026016, −4.20556242318142970415415468865, −3.79880476899905007401532986423, −3.01954207548583530131205558507, −2.41627675572756734882816938614, −1.51852708039212617380805070123, 0,
1.51852708039212617380805070123, 2.41627675572756734882816938614, 3.01954207548583530131205558507, 3.79880476899905007401532986423, 4.20556242318142970415415468865, 5.06850257247920738912499026016, 5.61017205144009625913946893858, 5.87134480744638375141626815412, 6.47166977500015585273964086632, 7.18964714691018416464928980947, 7.60840457104033872823456309121, 7.916186872483072797338932024160, 8.701941492535567968840109922622, 8.955462753706152130249428036919