| L(s) = 1 | + 2·4-s − 58·13-s − 60·16-s − 58·19-s + 38·25-s + 536·31-s + 166·37-s − 464·43-s − 116·52-s − 1.53e3·61-s − 248·64-s − 1.02e3·67-s − 274·73-s − 116·76-s − 950·79-s − 1.64e3·97-s + 76·100-s − 1.67e3·103-s + 436·109-s − 2.37e3·121-s + 1.07e3·124-s + 127-s + 131-s + 137-s + 139-s + 332·148-s + 149-s + ⋯ |
| L(s) = 1 | + 1/4·4-s − 1.23·13-s − 0.937·16-s − 0.700·19-s + 0.303·25-s + 3.10·31-s + 0.737·37-s − 1.64·43-s − 0.309·52-s − 3.21·61-s − 0.484·64-s − 1.86·67-s − 0.439·73-s − 0.175·76-s − 1.35·79-s − 1.71·97-s + 0.0759·100-s − 1.60·103-s + 0.383·109-s − 1.78·121-s + 0.776·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.184·148-s + 0.000549·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1750329 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1187270827\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1187270827\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 38 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2374 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 29 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 7234 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 29 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 17134 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 24950 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 268 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 83 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 64114 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 232 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 55294 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 204442 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 327526 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 767 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 511 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 207790 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 137 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 475 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 810646 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 1345138 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 821 T + p^{3} T^{2} )^{2} \) |
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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
−9.419602471758056801123093954848, −9.152316443089471794659179178093, −8.538635886343923013689920600592, −8.310705070985881812937405618986, −7.88377242399356307969112514146, −7.23455224651181555462884425304, −7.20161930688427556110873457732, −6.38866383018958728126462993261, −6.35149709707256741991171696085, −5.89763932988205537850467219808, −4.99242607032656423970397967889, −4.87523826574373259637822901004, −4.37092478035476485229913901925, −4.08709350189885043505217621106, −3.01748137964499579103483973215, −2.85386112303392669434071620745, −2.43117559333357534026084323746, −1.64034209015035595189846258349, −1.16324889426644068356962039458, −0.079910618417228206035140309057,
0.079910618417228206035140309057, 1.16324889426644068356962039458, 1.64034209015035595189846258349, 2.43117559333357534026084323746, 2.85386112303392669434071620745, 3.01748137964499579103483973215, 4.08709350189885043505217621106, 4.37092478035476485229913901925, 4.87523826574373259637822901004, 4.99242607032656423970397967889, 5.89763932988205537850467219808, 6.35149709707256741991171696085, 6.38866383018958728126462993261, 7.20161930688427556110873457732, 7.23455224651181555462884425304, 7.88377242399356307969112514146, 8.310705070985881812937405618986, 8.538635886343923013689920600592, 9.152316443089471794659179178093, 9.419602471758056801123093954848
