| L(s) = 1 | − 5-s + 7-s − 5·13-s − 7·17-s − 6·19-s − 23-s − 4·25-s + 29-s + 7·31-s − 35-s + 7·37-s − 12·41-s − 6·43-s + 7·47-s − 6·49-s − 2·53-s − 8·59-s − 10·61-s + 5·65-s + 9·67-s − 9·71-s + 8·73-s + 79-s − 14·83-s + 7·85-s + 12·89-s − 5·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.38·13-s − 1.69·17-s − 1.37·19-s − 0.208·23-s − 4/5·25-s + 0.185·29-s + 1.25·31-s − 0.169·35-s + 1.15·37-s − 1.87·41-s − 0.914·43-s + 1.02·47-s − 6/7·49-s − 0.274·53-s − 1.04·59-s − 1.28·61-s + 0.620·65-s + 1.09·67-s − 1.06·71-s + 0.936·73-s + 0.112·79-s − 1.53·83-s + 0.759·85-s + 1.27·89-s − 0.524·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 + 14 T + p T^{2} \) | 1.83.o |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 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
−13.23122284452804, −12.82161732204335, −12.35416476699718, −11.73989385832571, −11.57821472126824, −11.00702490127774, −10.46416266421264, −10.03011346922012, −9.614772913896198, −8.848411481263400, −8.653207331236493, −7.942897017866703, −7.734198146030456, −7.036417121861656, −6.442451278921638, −6.325116185541801, −5.432302451972758, −4.748427788286337, −4.506665560362473, −4.152568515284308, −3.269685320967104, −2.718385021312629, −2.020315740958778, −1.795495989408448, −0.5610044450055688, 0,
0.5610044450055688, 1.795495989408448, 2.020315740958778, 2.718385021312629, 3.269685320967104, 4.152568515284308, 4.506665560362473, 4.748427788286337, 5.432302451972758, 6.325116185541801, 6.442451278921638, 7.036417121861656, 7.734198146030456, 7.942897017866703, 8.653207331236493, 8.848411481263400, 9.614772913896198, 10.03011346922012, 10.46416266421264, 11.00702490127774, 11.57821472126824, 11.73989385832571, 12.35416476699718, 12.82161732204335, 13.23122284452804
