| L(s) = 1 | − 3·4-s − 18·7-s + 68·13-s − 55·16-s − 202·19-s + 54·28-s − 6·31-s − 134·37-s − 274·43-s − 443·49-s − 204·52-s − 1.12e3·61-s + 357·64-s + 2.08e3·67-s − 1.00e3·73-s + 606·76-s + 1.23e3·79-s − 1.22e3·91-s − 38·97-s + 298·103-s − 2.35e3·109-s + 990·112-s − 2.45e3·121-s + 18·124-s + 127-s + 131-s + 3.63e3·133-s + ⋯ |
| L(s) = 1 | − 3/8·4-s − 0.971·7-s + 1.45·13-s − 0.859·16-s − 2.43·19-s + 0.364·28-s − 0.0347·31-s − 0.595·37-s − 0.971·43-s − 1.29·49-s − 0.544·52-s − 2.36·61-s + 0.697·64-s + 3.80·67-s − 1.61·73-s + 0.914·76-s + 1.75·79-s − 1.41·91-s − 0.0397·97-s + 0.285·103-s − 2.06·109-s + 0.835·112-s − 1.84·121-s + 0.0130·124-s + 0.000698·127-s + 0.000666·131-s + 2.37·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 3 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 9 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2454 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 34 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 9774 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 101 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 12634 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 21270 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 67 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 97074 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 137 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 194334 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 161718 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 350574 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 563 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 1044 T + p^{3} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 + 98010 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 503 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 615 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 1086946 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 462238 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 19 T + p^{3} T^{2} )^{2} \) |
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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
−9.653615177128799339858546407925, −9.624252559188513452128432477543, −8.882389607184641958203843053462, −8.658644378013392503406004328059, −8.327029873538318416435291916222, −7.82661141542130539601932856314, −7.12933377947154105195811115838, −6.45137704185909663750377940568, −6.40266387985141217735072834023, −6.18676861023080942750058863334, −5.14299765028413885321342381311, −4.95084455493564077632182480853, −4.01316260823294156954640304447, −3.95009122690299283750359061872, −3.29679487459214804747284798364, −2.56876196431752541391331981678, −1.95046382304225445587043269107, −1.21980592044597267614797041905, 0, 0,
1.21980592044597267614797041905, 1.95046382304225445587043269107, 2.56876196431752541391331981678, 3.29679487459214804747284798364, 3.95009122690299283750359061872, 4.01316260823294156954640304447, 4.95084455493564077632182480853, 5.14299765028413885321342381311, 6.18676861023080942750058863334, 6.40266387985141217735072834023, 6.45137704185909663750377940568, 7.12933377947154105195811115838, 7.82661141542130539601932856314, 8.327029873538318416435291916222, 8.658644378013392503406004328059, 8.882389607184641958203843053462, 9.624252559188513452128432477543, 9.653615177128799339858546407925
