| L(s) = 1 | + 5-s + 7-s + 11-s − 13-s + 17-s + 5·19-s + 23-s − 4·25-s + 2·29-s − 6·31-s + 35-s + 8·37-s − 5·41-s − 43-s + 2·47-s + 49-s + 6·53-s + 55-s + 2·59-s + 8·61-s − 65-s − 4·67-s + 12·71-s + 14·73-s + 77-s − 10·79-s + 6·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.377·7-s + 0.301·11-s − 0.277·13-s + 0.242·17-s + 1.14·19-s + 0.208·23-s − 4/5·25-s + 0.371·29-s − 1.07·31-s + 0.169·35-s + 1.31·37-s − 0.780·41-s − 0.152·43-s + 0.291·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s + 0.260·59-s + 1.02·61-s − 0.124·65-s − 0.488·67-s + 1.42·71-s + 1.63·73-s + 0.113·77-s − 1.12·79-s + 0.658·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.511246944\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.511246944\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - T + p T^{2} \) | 1.23.ab |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 5 T + p T^{2} \) | 1.41.f |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 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
−7.79039512801415014044992382786, −7.10392357800598505289037498515, −6.43826974052610969397430295611, −5.52727847194738197229729450405, −5.22132539443336312033338184243, −4.22032454121895117601748302045, −3.51249240834977465257373983267, −2.57584654193292557466337067802, −1.75775551472930221542437964705, −0.797723128774920797158434020972,
0.797723128774920797158434020972, 1.75775551472930221542437964705, 2.57584654193292557466337067802, 3.51249240834977465257373983267, 4.22032454121895117601748302045, 5.22132539443336312033338184243, 5.52727847194738197229729450405, 6.43826974052610969397430295611, 7.10392357800598505289037498515, 7.79039512801415014044992382786
