| L(s) = 1 | − 2·4-s + 4·11-s + 6·13-s + 4·16-s + 5·17-s − 19-s + 8·23-s + 10·29-s − 31-s − 37-s + 3·41-s + 7·43-s − 8·44-s − 6·47-s − 7·49-s − 12·52-s + 5·53-s − 11·59-s − 12·61-s − 8·64-s + 2·67-s − 10·68-s − 9·71-s + 9·73-s + 2·76-s − 10·79-s + 9·83-s + ⋯ |
| L(s) = 1 | − 4-s + 1.20·11-s + 1.66·13-s + 16-s + 1.21·17-s − 0.229·19-s + 1.66·23-s + 1.85·29-s − 0.179·31-s − 0.164·37-s + 0.468·41-s + 1.06·43-s − 1.20·44-s − 0.875·47-s − 49-s − 1.66·52-s + 0.686·53-s − 1.43·59-s − 1.53·61-s − 64-s + 0.244·67-s − 1.21·68-s − 1.06·71-s + 1.05·73-s + 0.229·76-s − 1.12·79-s + 0.987·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6975 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.233719188\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.233719188\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 31 | \( 1 + T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 5 T + p T^{2} \) | 1.53.af |
| 59 | \( 1 + 11 T + p T^{2} \) | 1.59.l |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 9 T + p T^{2} \) | 1.73.aj |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−8.126841042351141845963997863794, −7.31097042399525277215034804174, −6.34007338486722898493151771356, −5.98251413708135890999710016088, −4.97575514056426673766295004172, −4.39439541315037019620696259522, −3.54078713550781629944050133520, −3.08674692778563903307254358785, −1.37034853526298486183491141757, −0.926959206925167676854426256809,
0.926959206925167676854426256809, 1.37034853526298486183491141757, 3.08674692778563903307254358785, 3.54078713550781629944050133520, 4.39439541315037019620696259522, 4.97575514056426673766295004172, 5.98251413708135890999710016088, 6.34007338486722898493151771356, 7.31097042399525277215034804174, 8.126841042351141845963997863794
