L(s) = 1 | + 3-s + 4·5-s + 2·7-s + 9-s + 5·13-s + 4·15-s − 3·17-s − 6·19-s + 2·21-s + 23-s + 2·25-s + 4·27-s − 29-s − 15·31-s + 8·35-s + 4·37-s + 5·39-s − 3·41-s + 4·43-s + 4·45-s − 5·47-s + 6·49-s − 3·51-s − 6·57-s + 8·59-s + 8·61-s + 2·63-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s + 0.755·7-s + 1/3·9-s + 1.38·13-s + 1.03·15-s − 0.727·17-s − 1.37·19-s + 0.436·21-s + 0.208·23-s + 2/5·25-s + 0.769·27-s − 0.185·29-s − 2.69·31-s + 1.35·35-s + 0.657·37-s + 0.800·39-s − 0.468·41-s + 0.609·43-s + 0.596·45-s − 0.729·47-s + 6/7·49-s − 0.420·51-s − 0.794·57-s + 1.04·59-s + 1.02·61-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100096 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100096 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.818491085\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.818491085\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
good | 3 | $C_2$$\times$$C_2$ | \( ( 1 - p T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 5 T + 16 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $D_{4}$ | \( 1 + T + 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 3 T + 40 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 5 T - 18 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $D_{4}$ | \( 1 - T + 24 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5 T + 68 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 12 T + 106 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T - 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 42 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−14.0965551300, −13.6548878091, −13.1009861061, −12.9207698074, −12.7486968851, −11.7209118095, −11.2005096550, −11.0155246327, −10.4282486380, −10.0356907258, −9.49396128896, −9.03349192179, −8.65462426011, −8.33808299572, −7.63851114103, −6.96614276936, −6.49901631082, −5.94769546268, −5.53221383172, −4.98888419012, −4.03423128040, −3.76739861417, −2.53994944446, −2.03129652045, −1.49141735514,
1.49141735514, 2.03129652045, 2.53994944446, 3.76739861417, 4.03423128040, 4.98888419012, 5.53221383172, 5.94769546268, 6.49901631082, 6.96614276936, 7.63851114103, 8.33808299572, 8.65462426011, 9.03349192179, 9.49396128896, 10.0356907258, 10.4282486380, 11.0155246327, 11.2005096550, 11.7209118095, 12.7486968851, 12.9207698074, 13.1009861061, 13.6548878091, 14.0965551300