| L(s) = 1 | + 2·5-s − 2·13-s + 2·17-s − 8·19-s − 25-s + 2·29-s + 8·31-s − 2·37-s − 10·41-s + 8·43-s + 8·47-s − 7·49-s + 2·53-s − 4·59-s − 2·61-s − 4·65-s + 8·67-s − 6·73-s + 8·79-s + 16·83-s + 4·85-s + 18·89-s − 16·95-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.554·13-s + 0.485·17-s − 1.83·19-s − 1/5·25-s + 0.371·29-s + 1.43·31-s − 0.328·37-s − 1.56·41-s + 1.21·43-s + 1.16·47-s − 49-s + 0.274·53-s − 0.520·59-s − 0.256·61-s − 0.496·65-s + 0.977·67-s − 0.702·73-s + 0.900·79-s + 1.75·83-s + 0.433·85-s + 1.90·89-s − 1.64·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76176 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76176 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.319617169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.319617169\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 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
−14.11105937925899, −13.44002806479497, −13.21672882128114, −12.54328336644026, −12.04499917835798, −11.76194886686988, −10.77140869157999, −10.55325352018327, −10.07902902061456, −9.536457548771704, −9.050876038703058, −8.468503906162457, −7.977150761833070, −7.420488360250268, −6.549663363714895, −6.416310952832961, −5.798505921163257, −5.099694721292941, −4.676759267632037, −3.999966271853978, −3.323799955977657, −2.457586977429502, −2.181666526052666, −1.388952331869960, −0.4923239858664202,
0.4923239858664202, 1.388952331869960, 2.181666526052666, 2.457586977429502, 3.323799955977657, 3.999966271853978, 4.676759267632037, 5.099694721292941, 5.798505921163257, 6.416310952832961, 6.549663363714895, 7.420488360250268, 7.977150761833070, 8.468503906162457, 9.050876038703058, 9.536457548771704, 10.07902902061456, 10.55325352018327, 10.77140869157999, 11.76194886686988, 12.04499917835798, 12.54328336644026, 13.21672882128114, 13.44002806479497, 14.11105937925899
