| L(s) = 1 | − 2·2-s − 2·3-s + 2·4-s − 4·5-s + 4·6-s + 2·7-s − 4·8-s + 3·9-s + 8·10-s − 4·12-s − 2·13-s − 4·14-s + 8·15-s + 8·16-s + 4·17-s − 6·18-s − 6·19-s − 8·20-s − 4·21-s + 10·23-s + 8·24-s + 5·25-s + 4·26-s − 4·27-s + 4·28-s − 6·29-s − 16·30-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s + 0.755·7-s − 1.41·8-s + 9-s + 2.52·10-s − 1.15·12-s − 0.554·13-s − 1.06·14-s + 2.06·15-s + 2·16-s + 0.970·17-s − 1.41·18-s − 1.37·19-s − 1.78·20-s − 0.872·21-s + 2.08·23-s + 1.63·24-s + 25-s + 0.784·26-s − 0.769·27-s + 0.755·28-s − 1.11·29-s − 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6456681 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 20 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 10 T + 68 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 64 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 66 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 6 T + 83 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 83 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 6 T + 104 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 6 T + 4 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 10 T + 168 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 162 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 199 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 139 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 10 T + 216 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
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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
−8.698260336979652135376052073130, −8.428423258237489954868377460935, −7.85556055085453594432530961538, −7.78804964737230667803590063104, −7.20676471684958324358432714130, −6.90251563466621336204177858694, −6.75073317584106230056355407392, −5.96508684095004937402906590214, −5.70351420201743719451858530118, −5.18556283625292989167181105770, −4.77512791348000032224381184044, −4.44395723461243615445312634453, −3.78587428998334391073947166421, −3.46785728088528978855883314847, −2.92746895862281881226893892614, −2.24330383644296391032755405416, −1.39636133902626387189181018692, −0.972105229939169897410053336497, 0, 0,
0.972105229939169897410053336497, 1.39636133902626387189181018692, 2.24330383644296391032755405416, 2.92746895862281881226893892614, 3.46785728088528978855883314847, 3.78587428998334391073947166421, 4.44395723461243615445312634453, 4.77512791348000032224381184044, 5.18556283625292989167181105770, 5.70351420201743719451858530118, 5.96508684095004937402906590214, 6.75073317584106230056355407392, 6.90251563466621336204177858694, 7.20676471684958324358432714130, 7.78804964737230667803590063104, 7.85556055085453594432530961538, 8.428423258237489954868377460935, 8.698260336979652135376052073130
