| L(s) = 1 | − 5-s − 9-s − 2·11-s − 2·31-s + 45-s − 2·49-s + 2·55-s − 2·59-s + 2·71-s + 2·89-s + 2·99-s + 3·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 5-s − 9-s − 2·11-s − 2·31-s + 45-s − 2·49-s + 2·55-s − 2·59-s + 2·71-s + 2·89-s + 2·99-s + 3·121-s + 125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 2·155-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12390400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2598556673\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2598556673\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.428504491463141679516085083156, −8.650368751680316502717473649803, −8.066362191758170309427956179372, −7.75819046476505591318378039814, −7.72157126202704508272933977698, −7.21934500546732208509083743260, −6.80635429246482385375611906903, −6.22527588521186774159283290465, −5.95882370993506645310466065945, −5.40530249119367340094434705633, −5.27416466078034725385693887564, −4.62107298996191751137655508229, −4.60176857958982066703390468480, −3.69501106849078729635575909196, −3.30190917640204802683499719936, −3.29353579526678005725077686489, −2.36846965196186028614873395685, −2.28477131878187930013813737081, −1.44321629673653656317330822670, −0.28948903540452816138240000563,
0.28948903540452816138240000563, 1.44321629673653656317330822670, 2.28477131878187930013813737081, 2.36846965196186028614873395685, 3.29353579526678005725077686489, 3.30190917640204802683499719936, 3.69501106849078729635575909196, 4.60176857958982066703390468480, 4.62107298996191751137655508229, 5.27416466078034725385693887564, 5.40530249119367340094434705633, 5.95882370993506645310466065945, 6.22527588521186774159283290465, 6.80635429246482385375611906903, 7.21934500546732208509083743260, 7.72157126202704508272933977698, 7.75819046476505591318378039814, 8.066362191758170309427956179372, 8.650368751680316502717473649803, 9.428504491463141679516085083156
