| L(s) = 1 | − 5-s − 7-s + 2·13-s + 4·17-s + 4·19-s − 23-s − 4·25-s − 10·29-s + 9·31-s + 35-s − 2·37-s + 3·41-s + 43-s − 4·47-s − 6·49-s − 5·53-s + 14·59-s − 10·61-s − 2·65-s + 14·67-s + 12·71-s + 8·73-s + 7·79-s + 6·83-s − 4·85-s − 11·89-s − 2·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s + 0.554·13-s + 0.970·17-s + 0.917·19-s − 0.208·23-s − 4/5·25-s − 1.85·29-s + 1.61·31-s + 0.169·35-s − 0.328·37-s + 0.468·41-s + 0.152·43-s − 0.583·47-s − 6/7·49-s − 0.686·53-s + 1.82·59-s − 1.28·61-s − 0.248·65-s + 1.71·67-s + 1.42·71-s + 0.936·73-s + 0.787·79-s + 0.658·83-s − 0.433·85-s − 1.16·89-s − 0.209·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.234261621\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.234261621\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 5 T + p T^{2} \) | 1.53.f |
| 59 | \( 1 - 14 T + p T^{2} \) | 1.59.ao |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 11 T + p T^{2} \) | 1.89.l |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.00041263436590, −12.59163477833016, −12.13360584518914, −11.57492517305844, −11.29341181782509, −10.85824849347939, −9.998205563386583, −9.846618715432049, −9.425751465601169, −8.773225350418974, −8.194562490404308, −7.788905147221794, −7.484440563162218, −6.788519458143159, −6.249754027116303, −5.862647081902823, −5.202482267832373, −4.848208011531501, −3.967568928273829, −3.572256183083467, −3.294898699950896, −2.442655184964730, −1.835827910897311, −1.055862018478154, −0.4797407370165758,
0.4797407370165758, 1.055862018478154, 1.835827910897311, 2.442655184964730, 3.294898699950896, 3.572256183083467, 3.967568928273829, 4.848208011531501, 5.202482267832373, 5.862647081902823, 6.249754027116303, 6.788519458143159, 7.484440563162218, 7.788905147221794, 8.194562490404308, 8.773225350418974, 9.425751465601169, 9.846618715432049, 9.998205563386583, 10.85824849347939, 11.29341181782509, 11.57492517305844, 12.13360584518914, 12.59163477833016, 13.00041263436590
