| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s − 11-s + 13-s + 16-s + 4·17-s − 6·19-s − 20-s + 22-s − 3·23-s − 4·25-s − 26-s + 5·29-s − 4·31-s − 32-s − 4·34-s + 10·37-s + 6·38-s + 40-s − 7·41-s − 5·43-s − 44-s + 3·46-s − 8·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.277·13-s + 1/4·16-s + 0.970·17-s − 1.37·19-s − 0.223·20-s + 0.213·22-s − 0.625·23-s − 4/5·25-s − 0.196·26-s + 0.928·29-s − 0.718·31-s − 0.176·32-s − 0.685·34-s + 1.64·37-s + 0.973·38-s + 0.158·40-s − 1.09·41-s − 0.762·43-s − 0.150·44-s + 0.442·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2761640258\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2761640258\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 + 5 T + p T^{2} \) | 1.43.f |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.51885150388327, −12.96217440504157, −12.44593362826787, −11.96498896562590, −11.62797135310278, −10.96994456113448, −10.62787306585885, −10.05950773305995, −9.715035050045743, −9.120412139914579, −8.473228288343919, −8.131867958964326, −7.772553455453789, −7.237364980073914, −6.450827796618966, −6.243113089158621, −5.576350625029440, −4.925059744181248, −4.258131884705031, −3.778862186362597, −3.079070392840519, −2.553838192028614, −1.736150400021743, −1.270664108448216, −0.1809808702367491,
0.1809808702367491, 1.270664108448216, 1.736150400021743, 2.553838192028614, 3.079070392840519, 3.778862186362597, 4.258131884705031, 4.925059744181248, 5.576350625029440, 6.243113089158621, 6.450827796618966, 7.237364980073914, 7.772553455453789, 8.131867958964326, 8.473228288343919, 9.120412139914579, 9.715035050045743, 10.05950773305995, 10.62787306585885, 10.96994456113448, 11.62797135310278, 11.96498896562590, 12.44593362826787, 12.96217440504157, 13.51885150388327
