| L(s) = 1 | − 2·4-s − 2·9-s − 12·11-s + 3·16-s + 20·19-s + 10·29-s − 12·31-s + 4·36-s − 2·41-s + 24·44-s + 20·49-s − 22·61-s − 4·64-s − 12·71-s − 40·76-s + 3·81-s − 10·89-s + 24·99-s + 38·101-s + 30·109-s − 20·116-s + 56·121-s + 24·124-s + 127-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 4-s − 2/3·9-s − 3.61·11-s + 3/4·16-s + 4.58·19-s + 1.85·29-s − 2.15·31-s + 2/3·36-s − 0.312·41-s + 3.61·44-s + 20/7·49-s − 2.81·61-s − 1/2·64-s − 1.42·71-s − 4.58·76-s + 1/3·81-s − 1.05·89-s + 2.41·99-s + 3.78·101-s + 2.87·109-s − 1.85·116-s + 5.09·121-s + 2.15·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3670052999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3670052999\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_4$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 25 T^{2} + 393 T^{4} - 25 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 5 T^{2} - 327 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 5 T + 63 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 65 T^{2} + 2433 T^{4} - 65 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 10078 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 80 T^{2} + 5038 T^{4} - 80 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 5 T^{2} + 4713 T^{4} - 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 160 T^{2} + 13758 T^{4} - 160 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_4$ | \( ( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 185 T^{2} + 17313 T^{4} - 185 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 130 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 5 T + 183 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 145 T^{2} + 19113 T^{4} - 145 p^{2} T^{6} + p^{4} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.80491862142006246606089942477, −5.78810828557661930102692055087, −5.51475481878090502954690199630, −5.48657873112552013723314203024, −5.30077673570284776819161951684, −5.01852125174198936136235690463, −4.87991577982101179458425478992, −4.63877189670983758577076197219, −4.63525716303569165153779198291, −4.35814978071044102626754047313, −3.86669965986909404876031540522, −3.77096261010516329178413532741, −3.32984324459851185681060694895, −3.23898963569154067218533385062, −3.16654381156014955891190649040, −2.93167212449551053568073219845, −2.74356135971675945942037262791, −2.52187506163899983044044301425, −2.23496479814391807421934615549, −1.77304568504698468588190871500, −1.70728067262587050533389121658, −1.08991769812352152234029935817, −0.815296902955333526087965922843, −0.71628230225356980410554219470, −0.11521953712764448075823316956,
0.11521953712764448075823316956, 0.71628230225356980410554219470, 0.815296902955333526087965922843, 1.08991769812352152234029935817, 1.70728067262587050533389121658, 1.77304568504698468588190871500, 2.23496479814391807421934615549, 2.52187506163899983044044301425, 2.74356135971675945942037262791, 2.93167212449551053568073219845, 3.16654381156014955891190649040, 3.23898963569154067218533385062, 3.32984324459851185681060694895, 3.77096261010516329178413532741, 3.86669965986909404876031540522, 4.35814978071044102626754047313, 4.63525716303569165153779198291, 4.63877189670983758577076197219, 4.87991577982101179458425478992, 5.01852125174198936136235690463, 5.30077673570284776819161951684, 5.48657873112552013723314203024, 5.51475481878090502954690199630, 5.78810828557661930102692055087, 5.80491862142006246606089942477
