| L(s) = 1 | + 6·7-s + 5·9-s − 6·17-s + 8·23-s + 9·25-s − 8·31-s + 24·41-s + 18·47-s + 13·49-s + 30·63-s − 30·71-s + 4·73-s + 16·79-s + 16·81-s − 4·89-s + 20·97-s + 8·103-s − 12·113-s − 36·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 30·153-s + ⋯ |
| L(s) = 1 | + 2.26·7-s + 5/3·9-s − 1.45·17-s + 1.66·23-s + 9/5·25-s − 1.43·31-s + 3.74·41-s + 2.62·47-s + 13/7·49-s + 3.77·63-s − 3.56·71-s + 0.468·73-s + 1.80·79-s + 16/9·81-s − 0.423·89-s + 2.03·97-s + 0.788·103-s − 1.12·113-s − 3.30·119-s + 1.63·121-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 − 2.42·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11075584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.763791777\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.763791777\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 13 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 9 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 49 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−8.880157672028371211683869038403, −8.628387802360297444039085334453, −7.905114463359413762411237583990, −7.50211611712680101994387131035, −7.35400998102095319157005823962, −7.32767881993696853970978007488, −6.49476624921280522772804785712, −6.42416732087092013598387706853, −5.56761019982169990166111603987, −5.38607867962064949987032330061, −4.80604899610561460037535656113, −4.60549786053395597259857643766, −4.17543957128242856597877921997, −4.15457340430832945190843238100, −3.24796225218268772921656850650, −2.63266241948440037720055765942, −2.19254625144669468143236401663, −1.76602891595524199945264321345, −1.04747734778876854886060885548, −0.944510207715478198284809024647,
0.944510207715478198284809024647, 1.04747734778876854886060885548, 1.76602891595524199945264321345, 2.19254625144669468143236401663, 2.63266241948440037720055765942, 3.24796225218268772921656850650, 4.15457340430832945190843238100, 4.17543957128242856597877921997, 4.60549786053395597259857643766, 4.80604899610561460037535656113, 5.38607867962064949987032330061, 5.56761019982169990166111603987, 6.42416732087092013598387706853, 6.49476624921280522772804785712, 7.32767881993696853970978007488, 7.35400998102095319157005823962, 7.50211611712680101994387131035, 7.905114463359413762411237583990, 8.628387802360297444039085334453, 8.880157672028371211683869038403
