| L(s) = 1 | − 2·2-s + 3·3-s + 3·4-s + 2·5-s − 6·6-s − 4·8-s + 6·9-s − 4·10-s − 11-s + 9·12-s − 6·13-s + 6·15-s + 5·16-s − 5·17-s − 12·18-s − 7·19-s + 6·20-s + 2·22-s − 4·23-s − 12·24-s + 5·25-s + 12·26-s + 9·27-s + 4·29-s − 12·30-s + 12·31-s − 6·32-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.73·3-s + 3/2·4-s + 0.894·5-s − 2.44·6-s − 1.41·8-s + 2·9-s − 1.26·10-s − 0.301·11-s + 2.59·12-s − 1.66·13-s + 1.54·15-s + 5/4·16-s − 1.21·17-s − 2.82·18-s − 1.60·19-s + 1.34·20-s + 0.426·22-s − 0.834·23-s − 2.44·24-s + 25-s + 2.35·26-s + 1.73·27-s + 0.742·29-s − 2.19·30-s + 2.15·31-s − 1.06·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.951480872\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.951480872\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 4 T - 13 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 3 T - 32 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 12 T + 91 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 16 T + 173 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−10.01774042577781044569141894288, −9.781906937207063259766060784799, −9.727441805925686140107129515604, −8.866934552999432882079680967780, −8.769191358265987266659673664024, −8.329195165711868322342342496364, −8.088511044148036834990943646146, −7.60830750679636519801628642651, −6.97781512180445837306400454561, −6.61547509958852745718702516373, −6.54013398236485940008217571986, −5.65972631198221324511713976871, −4.97042087332050589331086009998, −4.44007009865832714635654359256, −3.98163625808823220186171275637, −2.99637338889730586992485892775, −2.60537293457361060008247803849, −2.08504426574558671157875234302, −2.04445571257970849994995682780, −0.72886438361820211365616066580,
0.72886438361820211365616066580, 2.04445571257970849994995682780, 2.08504426574558671157875234302, 2.60537293457361060008247803849, 2.99637338889730586992485892775, 3.98163625808823220186171275637, 4.44007009865832714635654359256, 4.97042087332050589331086009998, 5.65972631198221324511713976871, 6.54013398236485940008217571986, 6.61547509958852745718702516373, 6.97781512180445837306400454561, 7.60830750679636519801628642651, 8.088511044148036834990943646146, 8.329195165711868322342342496364, 8.769191358265987266659673664024, 8.866934552999432882079680967780, 9.727441805925686140107129515604, 9.781906937207063259766060784799, 10.01774042577781044569141894288
