L(s) = 1 | − 3-s − 4·4-s + 5-s − 2·9-s − 5·11-s + 4·12-s + 13-s − 15-s + 12·16-s − 3·17-s − 4·20-s − 4·23-s − 6·25-s + 2·27-s + 4·29-s + 31-s + 5·33-s + 8·36-s − 39-s + 2·41-s − 12·43-s + 20·44-s − 2·45-s + 18·47-s − 12·48-s + 3·51-s − 4·52-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 2·4-s + 0.447·5-s − 2/3·9-s − 1.50·11-s + 1.15·12-s + 0.277·13-s − 0.258·15-s + 3·16-s − 0.727·17-s − 0.894·20-s − 0.834·23-s − 6/5·25-s + 0.384·27-s + 0.742·29-s + 0.179·31-s + 0.870·33-s + 4/3·36-s − 0.160·39-s + 0.312·41-s − 1.82·43-s + 3.01·44-s − 0.298·45-s + 2.62·47-s − 1.73·48-s + 0.420·51-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4036081 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 7 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - T )^{2} \) |
good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | $D_{4}$ | \( 1 + T + p T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - T + 7 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 5 T + 25 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - T + 23 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 7 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 49 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - T + 59 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 53 | $C_4$ | \( 1 - 9 T + 97 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 121 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 25 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 15 T + 199 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6 T - 41 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16 T + 217 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 181 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 15 T + 247 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
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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
−8.964553996192192304835834185381, −8.584337313748584171008274975841, −8.292461396959523451698842973673, −8.011964882236786287447004917189, −7.30765305099540558791090349400, −7.30630530579428335913130207460, −6.27324694550977427497958356907, −5.96693530228955912877585919749, −5.64135080796154896978556221998, −5.51138324868263757577110361822, −4.79040531856481382664727393853, −4.67960341094204373319430849009, −4.04935865377997381264276759061, −3.78605279638168857222091759104, −2.97327408335083045607922851207, −2.66115787201260131618437153258, −1.86748536113438001741414829712, −1.04438508788416614721215932322, 0, 0,
1.04438508788416614721215932322, 1.86748536113438001741414829712, 2.66115787201260131618437153258, 2.97327408335083045607922851207, 3.78605279638168857222091759104, 4.04935865377997381264276759061, 4.67960341094204373319430849009, 4.79040531856481382664727393853, 5.51138324868263757577110361822, 5.64135080796154896978556221998, 5.96693530228955912877585919749, 6.27324694550977427497958356907, 7.30630530579428335913130207460, 7.30765305099540558791090349400, 8.011964882236786287447004917189, 8.292461396959523451698842973673, 8.584337313748584171008274975841, 8.964553996192192304835834185381