| L(s) = 1 | − 3-s + 5-s + 7-s + 9-s + 11-s − 2·13-s − 15-s + 6·17-s + 4·19-s − 21-s + 25-s − 27-s − 6·29-s + 8·31-s − 33-s + 35-s + 10·37-s + 2·39-s − 6·41-s − 8·43-s + 45-s + 49-s − 6·51-s − 6·53-s + 55-s − 4·57-s − 12·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s + 0.917·19-s − 0.218·21-s + 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s + 0.169·35-s + 1.64·37-s + 0.320·39-s − 0.937·41-s − 1.21·43-s + 0.149·45-s + 1/7·49-s − 0.840·51-s − 0.824·53-s + 0.134·55-s − 0.529·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 73920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.530040801\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.530040801\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| good | 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 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 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.10318629356721, −13.53372255029989, −13.17125932920344, −12.39430906192806, −12.07995886982503, −11.65676123991096, −11.13596369554488, −10.57182163431833, −9.937286500083376, −9.641738521760384, −9.272753305979167, −8.306652462531926, −7.923124075734835, −7.425491166119153, −6.821891508656174, −6.173932636718767, −5.783950491642897, −5.073588309005020, −4.844385899905719, −4.038175654906841, −3.289538434496252, −2.794422687687344, −1.834525902136219, −1.306781798835225, −0.5805301393220787,
0.5805301393220787, 1.306781798835225, 1.834525902136219, 2.794422687687344, 3.289538434496252, 4.038175654906841, 4.844385899905719, 5.073588309005020, 5.783950491642897, 6.173932636718767, 6.821891508656174, 7.425491166119153, 7.923124075734835, 8.306652462531926, 9.272753305979167, 9.641738521760384, 9.937286500083376, 10.57182163431833, 11.13596369554488, 11.65676123991096, 12.07995886982503, 12.39430906192806, 13.17125932920344, 13.53372255029989, 14.10318629356721
