| L(s) = 1 | + 3·3-s − 5-s − 3·7-s + 6·9-s − 3·15-s + 2·17-s + 3·19-s − 9·21-s + 6·23-s − 4·25-s + 9·27-s + 29-s + 6·31-s + 3·35-s + 2·37-s + 7·41-s − 6·43-s − 6·45-s + 2·49-s + 6·51-s + 7·53-s + 9·57-s − 59-s + 10·61-s − 18·63-s + 12·67-s + 18·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s − 1.13·7-s + 2·9-s − 0.774·15-s + 0.485·17-s + 0.688·19-s − 1.96·21-s + 1.25·23-s − 4/5·25-s + 1.73·27-s + 0.185·29-s + 1.07·31-s + 0.507·35-s + 0.328·37-s + 1.09·41-s − 0.914·43-s − 0.894·45-s + 2/7·49-s + 0.840·51-s + 0.961·53-s + 1.19·57-s − 0.130·59-s + 1.28·61-s − 2.26·63-s + 1.46·67-s + 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.147305638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.147305638\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 59 | \( 1 + T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 3 T + p T^{2} \) | 1.19.ad |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + p T^{2} \) | 1.73.a |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−8.416589123403552810646653989818, −7.937175350036685947053462120055, −7.16832642589385521791701775871, −6.61054888890945159411278955230, −5.47776499479184401333616435379, −4.36701051923271712834895941141, −3.58228350128153383581553768141, −3.07734446264982855136715583460, −2.34001038667783991409650602290, −0.970785716438108711398341373331,
0.970785716438108711398341373331, 2.34001038667783991409650602290, 3.07734446264982855136715583460, 3.58228350128153383581553768141, 4.36701051923271712834895941141, 5.47776499479184401333616435379, 6.61054888890945159411278955230, 7.16832642589385521791701775871, 7.937175350036685947053462120055, 8.416589123403552810646653989818
