| L(s) = 1 | + 3-s − 5-s + 9-s + 5·11-s + 4·13-s − 15-s + 3·17-s + 19-s − 4·25-s + 27-s + 4·29-s + 2·31-s + 5·33-s − 4·37-s + 4·39-s + 4·41-s + 5·43-s − 45-s + 7·47-s + 3·51-s + 2·53-s − 5·55-s + 57-s − 10·59-s + 15·61-s − 4·65-s + 14·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.50·11-s + 1.10·13-s − 0.258·15-s + 0.727·17-s + 0.229·19-s − 4/5·25-s + 0.192·27-s + 0.742·29-s + 0.359·31-s + 0.870·33-s − 0.657·37-s + 0.640·39-s + 0.624·41-s + 0.762·43-s − 0.149·45-s + 1.02·47-s + 0.420·51-s + 0.274·53-s − 0.674·55-s + 0.132·57-s − 1.30·59-s + 1.92·61-s − 0.496·65-s + 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 178752 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.806984919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.806984919\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 15 T + p T^{2} \) | 1.61.ap |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−13.25528317376328, −12.57843159177353, −12.26669765063541, −11.76521600549929, −11.33542663166894, −10.90916048720326, −10.23893538915688, −9.798517684086011, −9.283039997445805, −8.897658654631538, −8.277922522249362, −8.116605387802151, −7.330619861339408, −6.965163942340555, −6.400216765687682, −5.882948054095060, −5.404473327213684, −4.555998509072018, −3.954796475999472, −3.828621545142971, −3.193376419970365, −2.544726061271224, −1.771073539807676, −1.159797567955047, −0.6892218160771766,
0.6892218160771766, 1.159797567955047, 1.771073539807676, 2.544726061271224, 3.193376419970365, 3.828621545142971, 3.954796475999472, 4.555998509072018, 5.404473327213684, 5.882948054095060, 6.400216765687682, 6.965163942340555, 7.330619861339408, 8.116605387802151, 8.277922522249362, 8.897658654631538, 9.283039997445805, 9.798517684086011, 10.23893538915688, 10.90916048720326, 11.33542663166894, 11.76521600549929, 12.26669765063541, 12.57843159177353, 13.25528317376328
