| L(s) = 1 | + 3-s + 7-s + 9-s + 4·11-s − 2·13-s − 2·17-s − 4·19-s + 21-s + 8·23-s + 27-s + 2·29-s + 4·33-s + 6·37-s − 2·39-s − 6·41-s − 4·43-s + 49-s − 2·51-s − 10·53-s − 4·57-s − 12·59-s − 14·61-s + 63-s − 12·67-s + 8·69-s − 8·71-s − 10·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.66·23-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s − 0.280·51-s − 1.37·53-s − 0.529·57-s − 1.56·59-s − 1.79·61-s + 0.125·63-s − 1.46·67-s + 0.963·69-s − 0.949·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.12573988369960, −14.73331558074438, −14.37222143593417, −13.65323976202345, −13.26239919281839, −12.67463436651081, −12.05875769121904, −11.66357432491662, −10.87769798750730, −10.62615860813420, −9.780123387905974, −9.178901980845782, −8.940690154649106, −8.315637066186162, −7.628001630033919, −7.148091403509354, −6.430792401510254, −6.133379824276299, −4.949674385328794, −4.653040935112325, −4.043026393831295, −3.173851334731726, −2.718590813178230, −1.728687980681014, −1.278343622431682, 0,
1.278343622431682, 1.728687980681014, 2.718590813178230, 3.173851334731726, 4.043026393831295, 4.653040935112325, 4.949674385328794, 6.133379824276299, 6.430792401510254, 7.148091403509354, 7.628001630033919, 8.315637066186162, 8.940690154649106, 9.178901980845782, 9.780123387905974, 10.62615860813420, 10.87769798750730, 11.66357432491662, 12.05875769121904, 12.67463436651081, 13.26239919281839, 13.65323976202345, 14.37222143593417, 14.73331558074438, 15.12573988369960
