| L(s) = 1 | − 4-s + 16-s − 2·25-s + 4·37-s − 64-s + 2·100-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s − 2·25-s + 4·37-s − 64-s + 2·100-s + 4·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s − 4·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8392881066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8392881066\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_1$ | \( ( 1 - T )^{4} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−9.677534016422554380889696489099, −9.246431659450565247006921931598, −9.087226457401992645957959057159, −8.405083282613219135238751943892, −8.136044192715390128572271088127, −7.79851068674101803897170280456, −7.46584590131062865531149361679, −7.04563518481578072468631546470, −6.14094826546581014578261410632, −6.10961685892133376495046241716, −5.81092166993369817941610123598, −5.14495973646711114181274785782, −4.64474357843096484590921649091, −4.40399428588893855454513683357, −3.81010106382271513261700290318, −3.58397151224920566203758156222, −2.75162345285640079406163651692, −2.36410989565923920772025707143, −1.54863206611834761541176199307, −0.74598289977922072572841224702,
0.74598289977922072572841224702, 1.54863206611834761541176199307, 2.36410989565923920772025707143, 2.75162345285640079406163651692, 3.58397151224920566203758156222, 3.81010106382271513261700290318, 4.40399428588893855454513683357, 4.64474357843096484590921649091, 5.14495973646711114181274785782, 5.81092166993369817941610123598, 6.10961685892133376495046241716, 6.14094826546581014578261410632, 7.04563518481578072468631546470, 7.46584590131062865531149361679, 7.79851068674101803897170280456, 8.136044192715390128572271088127, 8.405083282613219135238751943892, 9.087226457401992645957959057159, 9.246431659450565247006921931598, 9.677534016422554380889696489099
