L(s) = 1 | + 28·9-s + 280·17-s − 140·25-s − 728·41-s − 348·49-s + 3.64e3·73-s − 870·81-s − 2.18e3·89-s − 1.96e3·97-s − 3.64e3·113-s + 1.40e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 7.84e3·153-s + 157-s + 163-s + 167-s + 8.14e3·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
L(s) = 1 | + 1.03·9-s + 3.99·17-s − 1.11·25-s − 2.77·41-s − 1.01·49-s + 5.83·73-s − 1.19·81-s − 2.60·89-s − 2.05·97-s − 3.03·113-s + 1.05·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 4.14·153-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 3.70·169-s + 0.000439·173-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(3.711962442\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.711962442\) |
\(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
good | 3 | $C_2^2$ | \( ( 1 - 14 T^{2} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 + 14 p T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 174 T^{2} + p^{6} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 58 T + p^{3} T^{2} )^{2}( 1 + 58 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 4074 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 6958 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 754 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 33098 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + 39814 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 182 T + p^{3} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 141374 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 107294 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 282074 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 403998 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 399882 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 552526 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 703022 T^{2} + p^{6} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 910 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 + 525278 T^{2} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 632814 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 546 T + p^{3} T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 490 T + p^{3} T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.582775670188518468128407765530, −9.179098137008116716528552416196, −8.730562439661298931194605571401, −8.163832432087228483075589236271, −8.082688529071269249181523519771, −7.934803100557612782905712947515, −7.912089215880290909214454444923, −7.22580022671621646331627904796, −6.93904359422186065305981708579, −6.74195622579686793005980169405, −6.61216925073114069213018639094, −5.71317941436571232451105364763, −5.70107624936722437144644396843, −5.57790411167408496988752528083, −5.00896723109477689334745653313, −4.87596700180645501866596888215, −4.22586096781788846847025218758, −3.79679415022851530481763655912, −3.62644832231848533663856288426, −3.15706713652802246004115415058, −2.91788851574558275541648851809, −2.01542653512916390805880529757, −1.58060051165689216341964464967, −1.20462889427626375237706648641, −0.52240111283131630807705731875,
0.52240111283131630807705731875, 1.20462889427626375237706648641, 1.58060051165689216341964464967, 2.01542653512916390805880529757, 2.91788851574558275541648851809, 3.15706713652802246004115415058, 3.62644832231848533663856288426, 3.79679415022851530481763655912, 4.22586096781788846847025218758, 4.87596700180645501866596888215, 5.00896723109477689334745653313, 5.57790411167408496988752528083, 5.70107624936722437144644396843, 5.71317941436571232451105364763, 6.61216925073114069213018639094, 6.74195622579686793005980169405, 6.93904359422186065305981708579, 7.22580022671621646331627904796, 7.912089215880290909214454444923, 7.934803100557612782905712947515, 8.082688529071269249181523519771, 8.163832432087228483075589236271, 8.730562439661298931194605571401, 9.179098137008116716528552416196, 9.582775670188518468128407765530