| L(s) = 1 | − 2-s + 5·7-s − 8-s − 7·11-s + 3·13-s − 5·14-s − 16-s − 17-s + 7·22-s + 3·23-s − 3·26-s + 10·29-s + 2·31-s + 6·32-s + 34-s + 7·37-s + 7·41-s + 15·43-s − 3·46-s + 2·47-s + 8·49-s − 7·53-s − 5·56-s − 10·58-s + 11·59-s + 3·61-s − 2·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.88·7-s − 0.353·8-s − 2.11·11-s + 0.832·13-s − 1.33·14-s − 1/4·16-s − 0.242·17-s + 1.49·22-s + 0.625·23-s − 0.588·26-s + 1.85·29-s + 0.359·31-s + 1.06·32-s + 0.171·34-s + 1.15·37-s + 1.09·41-s + 2.28·43-s − 0.442·46-s + 0.291·47-s + 8/7·49-s − 0.961·53-s − 0.668·56-s − 1.31·58-s + 1.43·59-s + 0.384·61-s − 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.979838196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.979838196\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 5 T + 17 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 31 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 25 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + T + 31 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 3 T + 19 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 10 T + 70 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 7 T + 57 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7 T + 91 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 15 T + 139 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T + 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 7 T + 37 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 145 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 3 T + 121 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 125 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 193 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 169 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 2 T + 146 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−9.239237519441293871521255342653, −8.809206647398376497595043170181, −8.480453466489628367403068311486, −8.238573134005323466166479541866, −7.84134289604426828656476669863, −7.68366198725363219330179282207, −7.23474130489197555558553914900, −6.51159430598931659555008784983, −6.26742063818253214884215499158, −5.74617363931539656087311281631, −5.15194569520131981927564984877, −5.03931400311623492449024173076, −4.49282917038491840109831615849, −4.24039295146867036339107385486, −3.45763615800027137396861171501, −2.62915765805895904633698583374, −2.56327831467607934786761117001, −2.00647929830804394027415360413, −0.958443470869938674406594528062, −0.74851113578160851277905693506,
0.74851113578160851277905693506, 0.958443470869938674406594528062, 2.00647929830804394027415360413, 2.56327831467607934786761117001, 2.62915765805895904633698583374, 3.45763615800027137396861171501, 4.24039295146867036339107385486, 4.49282917038491840109831615849, 5.03931400311623492449024173076, 5.15194569520131981927564984877, 5.74617363931539656087311281631, 6.26742063818253214884215499158, 6.51159430598931659555008784983, 7.23474130489197555558553914900, 7.68366198725363219330179282207, 7.84134289604426828656476669863, 8.238573134005323466166479541866, 8.480453466489628367403068311486, 8.809206647398376497595043170181, 9.239237519441293871521255342653
