| L(s) = 1 | + 8·2-s + 48·4-s + 256·8-s + 1.28e3·16-s − 2.33e3·17-s − 4.48e3·19-s − 1.00e3·23-s + 7.71e3·31-s + 6.14e3·32-s − 1.86e4·34-s − 3.59e4·38-s − 8.00e3·46-s − 3.98e4·47-s − 3.33e4·49-s − 2.29e3·53-s − 7.63e4·61-s + 6.16e4·62-s + 2.86e4·64-s − 1.11e5·68-s − 2.15e5·76-s − 4.13e4·79-s − 1.93e5·83-s − 4.80e4·92-s − 3.18e5·94-s − 2.66e5·98-s − 1.83e4·106-s + 1.50e4·107-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s + 5/4·16-s − 1.95·17-s − 2.85·19-s − 0.394·23-s + 1.44·31-s + 1.06·32-s − 2.76·34-s − 4.03·38-s − 0.557·46-s − 2.62·47-s − 1.98·49-s − 0.112·53-s − 2.62·61-s + 2.03·62-s + 7/8·64-s − 2.93·68-s − 4.27·76-s − 0.745·79-s − 3.09·83-s − 0.591·92-s − 3.71·94-s − 2.80·98-s − 0.158·106-s + 0.126·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 33310 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 94074 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 463990 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 1166 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2244 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 500 T + p^{5} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40800682 T^{2} + p^{10} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3856 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 89804410 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 139752402 T^{2} + p^{10} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 265922422 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 19900 T + p^{5} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 1146 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 33744102 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 38158 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 1387552678 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3331783438 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 669338414 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 20664 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 96968 T + p^{5} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 7604090994 T^{2} + p^{10} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17140767490 T^{2} + p^{10} T^{4} \) |
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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
−10.18440690687814896798129073832, −9.818614998055517722764349907346, −9.061930879948732781601743855471, −8.619649579659139276781546456584, −8.143022479132229692421837456431, −7.85234456004308242208786877314, −6.79027163014327583528586683469, −6.73737122095056696438801894178, −6.23568701966129655737109488399, −6.02255528976815537226678959217, −4.98027790588367436545743788966, −4.74132969461172233351228165437, −4.18241215342490192383373159227, −4.01080699312534252667303566689, −2.84041712481054503178654387320, −2.79060703526177664011971312297, −1.71923132028723569310754104046, −1.70578637004818755124378382558, 0, 0,
1.70578637004818755124378382558, 1.71923132028723569310754104046, 2.79060703526177664011971312297, 2.84041712481054503178654387320, 4.01080699312534252667303566689, 4.18241215342490192383373159227, 4.74132969461172233351228165437, 4.98027790588367436545743788966, 6.02255528976815537226678959217, 6.23568701966129655737109488399, 6.73737122095056696438801894178, 6.79027163014327583528586683469, 7.85234456004308242208786877314, 8.143022479132229692421837456431, 8.619649579659139276781546456584, 9.061930879948732781601743855471, 9.818614998055517722764349907346, 10.18440690687814896798129073832
