L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s + 3·7-s − 8-s + 9-s + 4·10-s + 11-s − 12-s − 13-s − 3·14-s + 4·15-s + 16-s − 5·17-s − 18-s − 4·20-s − 3·21-s − 22-s + 2·23-s + 24-s + 4·25-s + 26-s − 4·27-s + 3·28-s − 5·29-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s + 1.03·15-s + 1/4·16-s − 1.21·17-s − 0.235·18-s − 0.894·20-s − 0.654·21-s − 0.213·22-s + 0.417·23-s + 0.204·24-s + 4/5·25-s + 0.196·26-s − 0.769·27-s + 0.566·28-s − 0.928·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 47 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 9 T + p T^{2} ) \) |
| 101 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 - 9 T + p T^{2} ) \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 5 | $D_{4}$ | \( 1 + 4 T + 12 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $D_{4}$ | \( 1 - T + 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + 5 T + 37 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 35 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 64 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 49 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 13 T + 107 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 116 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 144 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 7 T - 17 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + T - 75 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $D_{4}$ | \( 1 + 19 T + 201 T^{2} + 19 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6 T - 22 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 10 T + 84 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−15.4352041544, −14.8929193711, −14.5020315915, −14.1979767048, −13.1734373732, −12.7656622974, −12.4560681551, −11.6463966615, −11.3556346440, −11.1863056850, −10.9627026520, −10.1292087000, −9.33874594082, −9.07656426730, −8.43043742388, −7.73317274729, −7.54100008620, −7.25543094540, −6.29154009980, −5.74635361521, −4.89266784416, −4.16296785279, −3.98749340042, −2.70208615885, −1.55768390781, 0,
1.55768390781, 2.70208615885, 3.98749340042, 4.16296785279, 4.89266784416, 5.74635361521, 6.29154009980, 7.25543094540, 7.54100008620, 7.73317274729, 8.43043742388, 9.07656426730, 9.33874594082, 10.1292087000, 10.9627026520, 11.1863056850, 11.3556346440, 11.6463966615, 12.4560681551, 12.7656622974, 13.1734373732, 14.1979767048, 14.5020315915, 14.8929193711, 15.4352041544