| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 2·5-s + 4·6-s − 6·7-s − 4·8-s + 3·9-s − 4·10-s + 4·11-s − 6·12-s + 12·14-s − 4·15-s + 5·16-s − 8·17-s − 6·18-s + 8·19-s + 6·20-s + 12·21-s − 8·22-s + 8·24-s + 3·25-s − 4·27-s − 18·28-s − 4·29-s + 8·30-s − 4·31-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s − 2.26·7-s − 1.41·8-s + 9-s − 1.26·10-s + 1.20·11-s − 1.73·12-s + 3.20·14-s − 1.03·15-s + 5/4·16-s − 1.94·17-s − 1.41·18-s + 1.83·19-s + 1.34·20-s + 2.61·21-s − 1.70·22-s + 1.63·24-s + 3/5·25-s − 0.769·27-s − 3.40·28-s − 0.742·29-s + 1.46·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25704900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 13 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 23 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 51 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 6 T + 91 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 4 T + 107 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 170 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 98 T^{2} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 20 T + 210 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 190 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 35 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.86046389514128404549367068869, −7.77503294217511550215462595572, −6.99456315287783140265067756476, −6.96939823295719878145452420785, −6.57695750305345689497490233316, −6.52211409896226534399503654513, −5.96083419251577863744197504065, −5.87012850332408645937059775563, −5.15795260887058384132360691115, −5.12270288805622878232320633621, −4.08549948378650356597657732780, −4.02287765063027362291537088729, −3.26511790128494364400743485281, −3.14562861509133298524521506342, −2.28340704680664953639672420367, −2.17925659550521694591315289561, −1.18443507560774580709026840893, −1.13064255016672803054918077780, 0, 0,
1.13064255016672803054918077780, 1.18443507560774580709026840893, 2.17925659550521694591315289561, 2.28340704680664953639672420367, 3.14562861509133298524521506342, 3.26511790128494364400743485281, 4.02287765063027362291537088729, 4.08549948378650356597657732780, 5.12270288805622878232320633621, 5.15795260887058384132360691115, 5.87012850332408645937059775563, 5.96083419251577863744197504065, 6.52211409896226534399503654513, 6.57695750305345689497490233316, 6.96939823295719878145452420785, 6.99456315287783140265067756476, 7.77503294217511550215462595572, 7.86046389514128404549367068869
