| L(s) = 1 | + 3·5-s + 5·11-s − 13-s + 3·17-s + 19-s + 23-s + 4·25-s − 5·29-s − 8·31-s − 37-s − 12·41-s + 11·43-s − 8·47-s + 10·53-s + 15·55-s − 8·59-s + 13·61-s − 3·65-s + 2·67-s + 8·71-s − 11·73-s − 2·79-s − 18·83-s + 9·85-s + 18·89-s + 3·95-s − 10·97-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 1.50·11-s − 0.277·13-s + 0.727·17-s + 0.229·19-s + 0.208·23-s + 4/5·25-s − 0.928·29-s − 1.43·31-s − 0.164·37-s − 1.87·41-s + 1.67·43-s − 1.16·47-s + 1.37·53-s + 2.02·55-s − 1.04·59-s + 1.66·61-s − 0.372·65-s + 0.244·67-s + 0.949·71-s − 1.28·73-s − 0.225·79-s − 1.97·83-s + 0.976·85-s + 1.90·89-s + 0.307·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 366912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.449619734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.449619734\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 + 5 T + p T^{2} \) | 1.29.f |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 2 T + p T^{2} \) | 1.79.c |
| 83 | \( 1 + 18 T + p T^{2} \) | 1.83.s |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.59298952434072, −11.96207370568831, −11.68137016787542, −11.19807252409100, −10.61059923757246, −10.14272312815749, −9.772125942117018, −9.287586382122740, −9.087366830547316, −8.555187359029061, −7.942197683636216, −7.255396899120578, −6.984371493185211, −6.485354228110501, −5.929310644443987, −5.491595443463253, −5.270238150760113, −4.465407543906352, −3.932630778801224, −3.415448744171345, −2.939921281537873, −2.001282089259252, −1.856771329850706, −1.239120575643276, −0.5428131676093102,
0.5428131676093102, 1.239120575643276, 1.856771329850706, 2.001282089259252, 2.939921281537873, 3.415448744171345, 3.932630778801224, 4.465407543906352, 5.270238150760113, 5.491595443463253, 5.929310644443987, 6.485354228110501, 6.984371493185211, 7.255396899120578, 7.942197683636216, 8.555187359029061, 9.087366830547316, 9.287586382122740, 9.772125942117018, 10.14272312815749, 10.61059923757246, 11.19807252409100, 11.68137016787542, 11.96207370568831, 12.59298952434072
