| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s + 4·11-s + 12-s − 7·13-s + 16-s + 18-s − 2·19-s + 4·22-s + 6·23-s + 24-s − 5·25-s − 7·26-s + 27-s − 3·29-s − 6·31-s + 32-s + 4·33-s + 36-s + 6·37-s − 2·38-s − 7·39-s + 41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.288·12-s − 1.94·13-s + 1/4·16-s + 0.235·18-s − 0.458·19-s + 0.852·22-s + 1.25·23-s + 0.204·24-s − 25-s − 1.37·26-s + 0.192·27-s − 0.557·29-s − 1.07·31-s + 0.176·32-s + 0.696·33-s + 1/6·36-s + 0.986·37-s − 0.324·38-s − 1.12·39-s + 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 71094 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71094 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 - T \) | |
| 17 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 6 T + p T^{2} \) | 1.31.g |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 - 9 T + p T^{2} \) | 1.79.aj |
| 83 | \( 1 - 5 T + p T^{2} \) | 1.83.af |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.56123492719541, −14.04570627902216, −13.21178431742371, −13.03192356368577, −12.43519482284143, −11.97607761699636, −11.44811995106923, −11.05712784828348, −10.27371794847907, −9.716693790284230, −9.368182270164994, −8.917028663546205, −8.043654771081865, −7.691512573910268, −6.958924122564322, −6.804531258073361, −6.016606209228924, −5.351954246334322, −4.760372105731478, −4.366476895212232, −3.574260447010253, −3.252419110752290, −2.278809364979151, −2.044883685523202, −1.120995522892194, 0,
1.120995522892194, 2.044883685523202, 2.278809364979151, 3.252419110752290, 3.574260447010253, 4.366476895212232, 4.760372105731478, 5.351954246334322, 6.016606209228924, 6.804531258073361, 6.958924122564322, 7.691512573910268, 8.043654771081865, 8.917028663546205, 9.368182270164994, 9.716693790284230, 10.27371794847907, 11.05712784828348, 11.44811995106923, 11.97607761699636, 12.43519482284143, 13.03192356368577, 13.21178431742371, 14.04570627902216, 14.56123492719541
