| L(s) = 1 | + 3-s + 5-s + 3·7-s + 9-s + 11-s − 13-s + 15-s − 4·17-s − 5·19-s + 3·21-s − 4·23-s − 4·25-s + 27-s − 3·29-s − 2·31-s + 33-s + 3·35-s + 6·37-s − 39-s + 45-s − 3·47-s + 2·49-s − 4·51-s − 4·53-s + 55-s − 5·57-s + 4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.13·7-s + 1/3·9-s + 0.301·11-s − 0.277·13-s + 0.258·15-s − 0.970·17-s − 1.14·19-s + 0.654·21-s − 0.834·23-s − 4/5·25-s + 0.192·27-s − 0.557·29-s − 0.359·31-s + 0.174·33-s + 0.507·35-s + 0.986·37-s − 0.160·39-s + 0.149·45-s − 0.437·47-s + 2/7·49-s − 0.560·51-s − 0.549·53-s + 0.134·55-s − 0.662·57-s + 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.564818178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.564818178\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−12.68938821242512, −12.02107191721162, −11.58631576763089, −11.13275935240330, −10.85148969761201, −10.16889072725424, −9.806444588159825, −9.320741002101695, −8.880857562780953, −8.343578605444466, −8.066446911275005, −7.602814583638939, −7.025397966045835, −6.477510400268767, −6.105117942707480, −5.453537719445594, −4.999871787208356, −4.346444766081485, −4.096447129065397, −3.577612704755004, −2.620957333931795, −2.300377604033279, −1.784673424369709, −1.395845879077072, −0.3667141957012008,
0.3667141957012008, 1.395845879077072, 1.784673424369709, 2.300377604033279, 2.620957333931795, 3.577612704755004, 4.096447129065397, 4.346444766081485, 4.999871787208356, 5.453537719445594, 6.105117942707480, 6.477510400268767, 7.025397966045835, 7.602814583638939, 8.066446911275005, 8.343578605444466, 8.880857562780953, 9.320741002101695, 9.806444588159825, 10.16889072725424, 10.85148969761201, 11.13275935240330, 11.58631576763089, 12.02107191721162, 12.68938821242512
