| L(s) = 1 | + 2·3-s − 5-s − 4·7-s + 9-s + 11-s − 2·15-s + 17-s − 7·19-s − 8·21-s + 4·23-s + 25-s − 4·27-s − 4·29-s − 8·31-s + 2·33-s + 4·35-s − 11·37-s + 5·41-s + 11·43-s − 45-s − 11·47-s + 9·49-s + 2·51-s + 6·53-s − 55-s − 14·57-s − 12·59-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 0.516·15-s + 0.242·17-s − 1.60·19-s − 1.74·21-s + 0.834·23-s + 1/5·25-s − 0.769·27-s − 0.742·29-s − 1.43·31-s + 0.348·33-s + 0.676·35-s − 1.80·37-s + 0.780·41-s + 1.67·43-s − 0.149·45-s − 1.60·47-s + 9/7·49-s + 0.280·51-s + 0.824·53-s − 0.134·55-s − 1.85·57-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 148720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 - T + p T^{2} \) | 1.17.ab |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 11 T + p T^{2} \) | 1.37.l |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 11 T + p T^{2} \) | 1.43.al |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.85142169338500, −13.26458878389841, −12.89093835129232, −12.57726316531922, −12.16452570411847, −11.32344035408321, −10.95423624895272, −10.41344340335872, −9.903727250470795, −9.267447319557903, −8.990325757354418, −8.725938308906383, −8.074038478478722, −7.343096892085233, −7.208605789759468, −6.551507143717603, −5.923951664585513, −5.594142064533732, −4.595181063363161, −4.095706711658229, −3.587708980348864, −3.147260244631439, −2.734708937995099, −1.998681727658090, −1.372418174660617, 0, 0,
1.372418174660617, 1.998681727658090, 2.734708937995099, 3.147260244631439, 3.587708980348864, 4.095706711658229, 4.595181063363161, 5.594142064533732, 5.923951664585513, 6.551507143717603, 7.208605789759468, 7.343096892085233, 8.074038478478722, 8.725938308906383, 8.990325757354418, 9.267447319557903, 9.903727250470795, 10.41344340335872, 10.95423624895272, 11.32344035408321, 12.16452570411847, 12.57726316531922, 12.89093835129232, 13.26458878389841, 13.85142169338500
