| L(s) = 1 | − 2·2-s + 3·4-s − 4·8-s + 4·11-s + 5·16-s − 8·22-s + 12·23-s − 3·25-s + 10·29-s − 6·32-s + 6·37-s − 16·43-s + 12·44-s − 24·46-s + 6·50-s + 4·53-s − 20·58-s + 7·64-s − 4·67-s + 16·71-s − 12·74-s + 20·79-s + 32·86-s − 16·88-s + 36·92-s − 9·100-s − 8·106-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 1.41·8-s + 1.20·11-s + 5/4·16-s − 1.70·22-s + 2.50·23-s − 3/5·25-s + 1.85·29-s − 1.06·32-s + 0.986·37-s − 2.43·43-s + 1.80·44-s − 3.53·46-s + 0.848·50-s + 0.549·53-s − 2.62·58-s + 7/8·64-s − 0.488·67-s + 1.89·71-s − 1.39·74-s + 2.25·79-s + 3.45·86-s − 1.70·88-s + 3.75·92-s − 0.899·100-s − 0.777·106-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63011844 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.307023371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.307023371\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 115 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 82 T^{2} + p^{2} T^{4} \) |
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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
−8.135301477562809455737749899491, −7.86931557106097289841257454653, −7.20947190181953181761613286638, −6.90866100286798428839911175500, −6.87378310982354673955208919102, −6.45829022871101658517364508319, −6.19871338875784038582963629996, −5.69644365352156557125762753043, −5.14969909535983857629907450150, −5.03106029259959706479769588907, −4.41205278840699914032625597678, −4.16040627061869715929254423869, −3.35905857583578101133281409729, −3.32595408767676691465839263870, −2.84598264119518107197771829580, −2.34041846192201466129978557846, −1.81172171991786875001264397771, −1.42383395861123745096745604463, −0.76608166610878438855772466554, −0.66963094734333767848488414088,
0.66963094734333767848488414088, 0.76608166610878438855772466554, 1.42383395861123745096745604463, 1.81172171991786875001264397771, 2.34041846192201466129978557846, 2.84598264119518107197771829580, 3.32595408767676691465839263870, 3.35905857583578101133281409729, 4.16040627061869715929254423869, 4.41205278840699914032625597678, 5.03106029259959706479769588907, 5.14969909535983857629907450150, 5.69644365352156557125762753043, 6.19871338875784038582963629996, 6.45829022871101658517364508319, 6.87378310982354673955208919102, 6.90866100286798428839911175500, 7.20947190181953181761613286638, 7.86931557106097289841257454653, 8.135301477562809455737749899491
