| L(s) = 1 | − 4·3-s + 7·9-s − 28·13-s − 16·19-s + 5·25-s + 8·27-s + 62·31-s − 56·37-s + 112·39-s − 104·43-s + 64·57-s − 28·61-s − 8·67-s − 196·73-s − 20·75-s + 202·79-s − 95·81-s − 248·93-s + 26·97-s − 316·103-s + 16·109-s + 224·111-s − 196·117-s + 197·121-s + 127-s + 416·129-s + 131-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 7/9·9-s − 2.15·13-s − 0.842·19-s + 1/5·25-s + 8/27·27-s + 2·31-s − 1.51·37-s + 2.87·39-s − 2.41·43-s + 1.12·57-s − 0.459·61-s − 0.119·67-s − 2.68·73-s − 0.266·75-s + 2.55·79-s − 1.17·81-s − 8/3·93-s + 0.268·97-s − 3.06·103-s + 0.146·109-s + 2.01·111-s − 1.67·117-s + 1.62·121-s + 0.00787·127-s + 3.22·129-s + 0.00763·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345744 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2074650060\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2074650060\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + 4 T + p^{2} T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 197 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 142 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 44 T + p^{2} T^{2} )( 1 + 44 T + p^{2} T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 523 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 28 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 1138 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 52 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2798 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 5573 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6557 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 14 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8462 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 98 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 101 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 6173 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 11342 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p^{2} T^{2} )^{2} \) |
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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
−10.82814490971174592281727807401, −10.21224849833442047387093080856, −9.906590714202482069088939543627, −9.754279729321019567060758048347, −8.934998271972651614091197447185, −8.426943647766531072927289793921, −8.166884278255222912740524296203, −7.24296092559417277538933552567, −7.21820112608773065754999210266, −6.51812577972347896147062893766, −6.29118861602379451829732844858, −5.63055846787275580477636417100, −4.98883866185677708130807945783, −4.84669662174245123723607980174, −4.43098373758325541655009719846, −3.51010885759090636228886126646, −2.80778449839879205435168369357, −2.20164880660860203509727839033, −1.31374718233180899021946460016, −0.19406031293199586909545254811,
0.19406031293199586909545254811, 1.31374718233180899021946460016, 2.20164880660860203509727839033, 2.80778449839879205435168369357, 3.51010885759090636228886126646, 4.43098373758325541655009719846, 4.84669662174245123723607980174, 4.98883866185677708130807945783, 5.63055846787275580477636417100, 6.29118861602379451829732844858, 6.51812577972347896147062893766, 7.21820112608773065754999210266, 7.24296092559417277538933552567, 8.166884278255222912740524296203, 8.426943647766531072927289793921, 8.934998271972651614091197447185, 9.754279729321019567060758048347, 9.906590714202482069088939543627, 10.21224849833442047387093080856, 10.82814490971174592281727807401
