| L(s) = 1 | − 6·3-s + 21·9-s + 14·17-s + 14·23-s − 25-s − 54·27-s − 10·29-s + 18·43-s + 5·49-s − 84·51-s − 12·53-s − 10·61-s − 84·69-s + 6·75-s − 16·79-s + 108·81-s + 60·87-s + 18·101-s + 32·103-s − 6·107-s + 26·113-s + 13·121-s + 127-s − 108·129-s + 131-s + 137-s + 139-s + ⋯ |
| L(s) = 1 | − 3.46·3-s + 7·9-s + 3.39·17-s + 2.91·23-s − 1/5·25-s − 10.3·27-s − 1.85·29-s + 2.74·43-s + 5/7·49-s − 11.7·51-s − 1.64·53-s − 1.28·61-s − 10.1·69-s + 0.692·75-s − 1.80·79-s + 12·81-s + 6.43·87-s + 1.79·101-s + 3.15·103-s − 0.580·107-s + 2.44·113-s + 1.18·121-s + 0.0887·127-s − 9.50·129-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11424400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.013848256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.013848256\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2$ | \( ( 1 + p T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 37 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 65 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 33 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 93 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 133 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 129 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 73 T^{2} + p^{2} T^{4} \) |
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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
−9.043851085202135989293447394655, −8.313575066734613070156840536951, −7.68478546185792904855775722537, −7.35594269230784983402777603250, −7.23444124905394986400759370423, −7.07369675414016349169216927033, −6.19640453111125923135565079668, −6.04014989213173529271941018494, −5.67378014174418122355350416254, −5.65484072617136688869015866072, −5.09963019013441882214160535541, −4.80746468223207447065954966798, −4.50739656012929874061090337443, −3.91866914276414711119124742205, −3.23339388130858337175357263748, −3.11246541908254669871321818838, −1.88169768979696503699066199838, −1.30561566825963211679558134676, −0.864693575847817334454316249255, −0.59498162436130934325145437729,
0.59498162436130934325145437729, 0.864693575847817334454316249255, 1.30561566825963211679558134676, 1.88169768979696503699066199838, 3.11246541908254669871321818838, 3.23339388130858337175357263748, 3.91866914276414711119124742205, 4.50739656012929874061090337443, 4.80746468223207447065954966798, 5.09963019013441882214160535541, 5.65484072617136688869015866072, 5.67378014174418122355350416254, 6.04014989213173529271941018494, 6.19640453111125923135565079668, 7.07369675414016349169216927033, 7.23444124905394986400759370423, 7.35594269230784983402777603250, 7.68478546185792904855775722537, 8.313575066734613070156840536951, 9.043851085202135989293447394655
