| L(s) = 1 | − 7-s − 4·11-s + 2·13-s + 6·17-s − 6·19-s + 2·23-s + 6·29-s − 2·31-s + 8·37-s − 12·41-s + 4·43-s + 8·47-s + 49-s + 2·53-s − 12·59-s − 10·61-s − 4·67-s − 8·71-s + 2·73-s + 4·77-s − 8·79-s + 12·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 1.20·11-s + 0.554·13-s + 1.45·17-s − 1.37·19-s + 0.417·23-s + 1.11·29-s − 0.359·31-s + 1.31·37-s − 1.87·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 1.56·59-s − 1.28·61-s − 0.488·67-s − 0.949·71-s + 0.234·73-s + 0.455·77-s − 0.900·79-s + 1.27·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.666052916\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.666052916\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| good | 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 12 T + p T^{2} \) | 1.41.m |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−14.54887079298177, −13.89832548715059, −13.49495746952658, −12.95688787331375, −12.49501633493811, −12.10227278789913, −11.40200361905144, −10.74627682856197, −10.34399396251793, −10.09000227904657, −9.217754463004355, −8.787694330804353, −8.142968093692271, −7.731372706252597, −7.168964131592757, −6.404768328436767, −5.954710987580833, −5.447016305988081, −4.689601751801931, −4.229746495214898, −3.274732946537222, −2.975773437939070, −2.184154170314011, −1.331426654487773, −0.4676482437480996,
0.4676482437480996, 1.331426654487773, 2.184154170314011, 2.975773437939070, 3.274732946537222, 4.229746495214898, 4.689601751801931, 5.447016305988081, 5.954710987580833, 6.404768328436767, 7.168964131592757, 7.731372706252597, 8.142968093692271, 8.787694330804353, 9.217754463004355, 10.09000227904657, 10.34399396251793, 10.74627682856197, 11.40200361905144, 12.10227278789913, 12.49501633493811, 12.95688787331375, 13.49495746952658, 13.89832548715059, 14.54887079298177
