L(s) = 1 | − 2·2-s − 3-s + 3·4-s − 2·5-s + 2·6-s − 2·7-s − 4·8-s − 2·9-s + 4·10-s + 2·11-s − 3·12-s + 3·13-s + 4·14-s + 2·15-s + 5·16-s + 4·18-s − 19-s − 6·20-s + 2·21-s − 4·22-s + 4·23-s + 4·24-s + 3·25-s − 6·26-s + 2·27-s − 6·28-s + 2·29-s + ⋯ |
L(s) = 1 | − 1.41·2-s − 0.577·3-s + 3/2·4-s − 0.894·5-s + 0.816·6-s − 0.755·7-s − 1.41·8-s − 2/3·9-s + 1.26·10-s + 0.603·11-s − 0.866·12-s + 0.832·13-s + 1.06·14-s + 0.516·15-s + 5/4·16-s + 0.942·18-s − 0.229·19-s − 1.34·20-s + 0.436·21-s − 0.852·22-s + 0.834·23-s + 0.816·24-s + 3/5·25-s − 1.17·26-s + 0.384·27-s − 1.13·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64480900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64480900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 73 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 + T + 35 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_4$ | \( 1 - 5 T + 89 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 3 T + 15 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 11 T + 145 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 19 T + 3 p T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 3 T + 107 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T - 11 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 10 T + 170 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 11 T + 115 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 15 T + 205 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 178 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−7.57849412230644438874709624996, −7.50249723346780485371810207084, −6.89853018429893004459818117932, −6.86888593223588185077031334826, −6.19842344760944667264276275740, −6.12255971020768663035539228400, −5.80779368048224043108235005598, −5.44686298295976496887753624692, −4.84486118281074792464232892902, −4.36982150092787651795862067466, −4.08231269988424588437417230115, −3.62351440832817977415592038951, −3.03570118310711352692292604279, −3.02039811930809819720689560280, −2.44472297310702251366148272379, −1.83007173452350223987837138534, −1.07582882335504405277243524419, −1.03367787437003878904427192716, 0, 0,
1.03367787437003878904427192716, 1.07582882335504405277243524419, 1.83007173452350223987837138534, 2.44472297310702251366148272379, 3.02039811930809819720689560280, 3.03570118310711352692292604279, 3.62351440832817977415592038951, 4.08231269988424588437417230115, 4.36982150092787651795862067466, 4.84486118281074792464232892902, 5.44686298295976496887753624692, 5.80779368048224043108235005598, 6.12255971020768663035539228400, 6.19842344760944667264276275740, 6.86888593223588185077031334826, 6.89853018429893004459818117932, 7.50249723346780485371810207084, 7.57849412230644438874709624996