L(s) = 1 | − 8·4-s − 18·9-s + 6·11-s + 48·16-s + 38·19-s + 144·36-s − 48·44-s + 73·49-s + 206·61-s − 256·64-s − 304·76-s + 243·81-s − 108·99-s − 204·101-s − 215·121-s + 127-s + 131-s + 137-s + 139-s − 864·144-s + 149-s + 151-s + 157-s + 163-s + 167-s − 338·169-s − 684·171-s + ⋯ |
L(s) = 1 | − 2·4-s − 2·9-s + 6/11·11-s + 3·16-s + 2·19-s + 4·36-s − 1.09·44-s + 1.48·49-s + 3.37·61-s − 4·64-s − 4·76-s + 3·81-s − 1.09·99-s − 2.01·101-s − 1.77·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s − 6·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s − 2·169-s − 4·171-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9439256316\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9439256316\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 73 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 353 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 158 T^{2} + p^{4} T^{4} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 3527 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 1207 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 103 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 10033 T^{2} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 5678 T^{2} + p^{4} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
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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
−10.96348772132453140814658003016, −10.55417973163687070222589555231, −9.898551259455338163206544257748, −9.550768301795540069324048842231, −9.237904884078837185636814820836, −8.832656450362261840594529375692, −8.260771632350344332623621372625, −8.249656551299161358940077699160, −7.52347679468063298774791178230, −6.94912980040000317484386554876, −6.12877410451595404205008172589, −5.64315199029482536239989573967, −5.25321551886673374653250111097, −5.08250906432001103913579137753, −4.11342820987329293886349479990, −3.74022800459851719505078817882, −3.18221161227009083577586528517, −2.54783407496668986615495354907, −1.13332464440955089542180359985, −0.49320274841841272145180973859,
0.49320274841841272145180973859, 1.13332464440955089542180359985, 2.54783407496668986615495354907, 3.18221161227009083577586528517, 3.74022800459851719505078817882, 4.11342820987329293886349479990, 5.08250906432001103913579137753, 5.25321551886673374653250111097, 5.64315199029482536239989573967, 6.12877410451595404205008172589, 6.94912980040000317484386554876, 7.52347679468063298774791178230, 8.249656551299161358940077699160, 8.260771632350344332623621372625, 8.832656450362261840594529375692, 9.237904884078837185636814820836, 9.550768301795540069324048842231, 9.898551259455338163206544257748, 10.55417973163687070222589555231, 10.96348772132453140814658003016