| L(s) = 1 | + 7-s + 7·13-s − 19-s − 7·31-s − 11·37-s + 13·43-s − 6·49-s + 14·61-s + 16·67-s + 10·73-s − 13·79-s + 7·91-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 1.94·13-s − 0.229·19-s − 1.25·31-s − 1.80·37-s + 1.98·43-s − 6/7·49-s + 1.79·61-s + 1.95·67-s + 1.17·73-s − 1.46·79-s + 0.733·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.608736714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.608736714\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 7 T + p T^{2} \) | 1.13.ah |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−15.63840692885065, −14.79977652316173, −14.25383495309003, −13.89557055114073, −13.23125927465390, −12.75613564164663, −12.25059622083375, −11.42311893060763, −11.00117061010354, −10.75260882856203, −9.917680032951121, −9.298205866329219, −8.587821388840043, −8.409491412680172, −7.627927328672917, −6.908582754703935, −6.434800322090333, −5.592292523533869, −5.345119825256800, −4.285444635136744, −3.797021161345111, −3.223648458676837, −2.173915826117937, −1.528185746232112, −0.6708615971258727,
0.6708615971258727, 1.528185746232112, 2.173915826117937, 3.223648458676837, 3.797021161345111, 4.285444635136744, 5.345119825256800, 5.592292523533869, 6.434800322090333, 6.908582754703935, 7.627927328672917, 8.409491412680172, 8.587821388840043, 9.298205866329219, 9.917680032951121, 10.75260882856203, 11.00117061010354, 11.42311893060763, 12.25059622083375, 12.75613564164663, 13.23125927465390, 13.89557055114073, 14.25383495309003, 14.79977652316173, 15.63840692885065
