| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 3·7-s − 8-s + 9-s − 3·11-s − 12-s − 4·13-s + 3·14-s + 16-s − 17-s − 18-s − 5·19-s + 3·21-s + 3·22-s − 4·23-s + 24-s + 4·26-s − 27-s − 3·28-s + 7·31-s − 32-s + 3·33-s + 34-s + 36-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s − 1.10·13-s + 0.801·14-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 1.14·19-s + 0.654·21-s + 0.639·22-s − 0.834·23-s + 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.566·28-s + 1.25·31-s − 0.176·32-s + 0.522·33-s + 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3703155389\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3703155389\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 11 T + p T^{2} \) | 1.53.al |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 13 T + p T^{2} \) | 1.67.n |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.961979392039539567103663650957, −8.128954258469383063983693346653, −7.36152616094781964711149979371, −6.60909249458524641418646646362, −6.03504743133314897206509044898, −5.08721258126832127574248751642, −4.12994182228145702051862029277, −2.90365026445617287214084227965, −2.12091736431276150561568571367, −0.40978224747320534667331708449,
0.40978224747320534667331708449, 2.12091736431276150561568571367, 2.90365026445617287214084227965, 4.12994182228145702051862029277, 5.08721258126832127574248751642, 6.03504743133314897206509044898, 6.60909249458524641418646646362, 7.36152616094781964711149979371, 8.128954258469383063983693346653, 8.961979392039539567103663650957
