| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 4·19-s − 2·25-s − 6·32-s − 8·38-s − 2·49-s + 4·50-s + 7·64-s + 12·76-s − 81-s + 4·98-s − 6·100-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·152-s + 157-s + 2·162-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 5·16-s + 4·19-s − 2·25-s − 6·32-s − 8·38-s − 2·49-s + 4·50-s + 7·64-s + 12·76-s − 81-s + 4·98-s − 6·100-s + 127-s − 8·128-s + 131-s + 137-s + 139-s + 149-s + 151-s − 16·152-s + 157-s + 2·162-s + 163-s + 167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5345344 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5261019259\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5261019259\) |
| \(L(1)\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 17 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 5 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + T^{4} \) |
| 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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + T^{4} \) |
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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
−9.614264772597664196256758051938, −9.102005453951643520433010899772, −8.707458078458520920068480075152, −8.031326241540536464289443557890, −7.953517020825544983906502758866, −7.48956473704374004536494950397, −7.45270596970687120752617307169, −6.83830519716989257265969354464, −6.49323256411014365129778835052, −5.97865340789028463133336847762, −5.46389556050631479372512015207, −5.43953426940813327881853049335, −4.74657607714187197422294611105, −3.71682493019203716489889266243, −3.57168116561604662871776414935, −2.85772248277175471962597874792, −2.74398596629721884571899181962, −1.64456328964810622575787843166, −1.58160263771941075541406582282, −0.72067421084153552208929433333,
0.72067421084153552208929433333, 1.58160263771941075541406582282, 1.64456328964810622575787843166, 2.74398596629721884571899181962, 2.85772248277175471962597874792, 3.57168116561604662871776414935, 3.71682493019203716489889266243, 4.74657607714187197422294611105, 5.43953426940813327881853049335, 5.46389556050631479372512015207, 5.97865340789028463133336847762, 6.49323256411014365129778835052, 6.83830519716989257265969354464, 7.45270596970687120752617307169, 7.48956473704374004536494950397, 7.953517020825544983906502758866, 8.031326241540536464289443557890, 8.707458078458520920068480075152, 9.102005453951643520433010899772, 9.614264772597664196256758051938
