| L(s) = 1 | − 3-s + 5-s + 9-s − 3·11-s − 6·13-s − 15-s − 3·17-s + 19-s − 4·23-s − 4·25-s − 27-s + 10·29-s + 2·31-s + 3·33-s − 8·37-s + 6·39-s + 8·41-s − 43-s + 45-s + 3·47-s + 3·51-s + 6·53-s − 3·55-s − 57-s + 7·61-s − 6·65-s + 8·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.904·11-s − 1.66·13-s − 0.258·15-s − 0.727·17-s + 0.229·19-s − 0.834·23-s − 4/5·25-s − 0.192·27-s + 1.85·29-s + 0.359·31-s + 0.522·33-s − 1.31·37-s + 0.960·39-s + 1.24·41-s − 0.152·43-s + 0.149·45-s + 0.437·47-s + 0.420·51-s + 0.824·53-s − 0.404·55-s − 0.132·57-s + 0.896·61-s − 0.744·65-s + 0.977·67-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
| \(L(1)\) |
\(\approx\) |
\(0.7837564092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7837564092\) |
| \(L(\frac{3}{2})\) |
|
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.ab |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 8 T + p T^{2} \) | 1.41.ai |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.15079524006871, −12.55102403643197, −12.27682750809823, −11.80241953576681, −11.39363509095034, −10.61843638291023, −10.34338533052674, −9.950306224947892, −9.549431529818143, −8.941572385499926, −8.257401950141263, −7.922081146800026, −7.268500207937629, −6.879630224677085, −6.389651767454789, −5.657661283232425, −5.429454720347501, −4.765518148688046, −4.442242394543961, −3.764032658044037, −2.872292427221908, −2.384854264072437, −2.048944585893270, −1.075325093218620, −0.2806348541922004,
0.2806348541922004, 1.075325093218620, 2.048944585893270, 2.384854264072437, 2.872292427221908, 3.764032658044037, 4.442242394543961, 4.765518148688046, 5.429454720347501, 5.657661283232425, 6.389651767454789, 6.879630224677085, 7.268500207937629, 7.922081146800026, 8.257401950141263, 8.941572385499926, 9.549431529818143, 9.950306224947892, 10.34338533052674, 10.61843638291023, 11.39363509095034, 11.80241953576681, 12.27682750809823, 12.55102403643197, 13.15079524006871
