| L(s) = 1 | − 3-s + 7-s + 9-s − 2·17-s − 21-s + 4·23-s − 27-s + 2·29-s + 8·31-s − 8·37-s − 4·41-s − 8·43-s + 49-s + 2·51-s − 6·53-s − 8·59-s − 10·61-s + 63-s − 8·67-s − 4·69-s + 4·73-s + 12·79-s + 81-s + 8·83-s − 2·87-s − 12·89-s − 8·93-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.485·17-s − 0.218·21-s + 0.834·23-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 1.31·37-s − 0.624·41-s − 1.21·43-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 1.04·59-s − 1.28·61-s + 0.125·63-s − 0.977·67-s − 0.481·69-s + 0.468·73-s + 1.35·79-s + 1/9·81-s + 0.878·83-s − 0.214·87-s − 1.27·89-s − 0.829·93-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 354900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 354900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9589987474\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9589987474\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 - 12 T + p T^{2} \) | 1.79.am |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−12.31577221210085, −12.15553572839937, −11.63972366608532, −11.19495502491978, −10.62261643861264, −10.53160903564042, −9.807116138423727, −9.454926263781373, −8.780557429446098, −8.521055305923556, −7.874632387026893, −7.538483241249733, −6.774713121187291, −6.564851656924280, −6.158737336111705, −5.266667185556335, −5.158486473347562, −4.552217900644059, −4.168339694811939, −3.351684470439409, −2.982234128829571, −2.280601814141625, −1.537761340304248, −1.212439839870713, −0.2710249865740243,
0.2710249865740243, 1.212439839870713, 1.537761340304248, 2.280601814141625, 2.982234128829571, 3.351684470439409, 4.168339694811939, 4.552217900644059, 5.158486473347562, 5.266667185556335, 6.158737336111705, 6.564851656924280, 6.774713121187291, 7.538483241249733, 7.874632387026893, 8.521055305923556, 8.780557429446098, 9.454926263781373, 9.807116138423727, 10.53160903564042, 10.62261643861264, 11.19495502491978, 11.63972366608532, 12.15553572839937, 12.31577221210085
