| L(s) = 1 | + 3-s − 4·5-s − 2·9-s − 3·11-s − 5·13-s − 4·15-s − 17-s + 6·19-s − 4·23-s + 11·25-s − 5·27-s + 8·29-s − 3·33-s − 8·37-s − 5·39-s − 8·41-s + 10·43-s + 8·45-s − 2·47-s − 51-s + 3·53-s + 12·55-s + 6·57-s + 2·59-s − 8·61-s + 20·65-s + 14·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 2/3·9-s − 0.904·11-s − 1.38·13-s − 1.03·15-s − 0.242·17-s + 1.37·19-s − 0.834·23-s + 11/5·25-s − 0.962·27-s + 1.48·29-s − 0.522·33-s − 1.31·37-s − 0.800·39-s − 1.24·41-s + 1.52·43-s + 1.19·45-s − 0.291·47-s − 0.140·51-s + 0.412·53-s + 1.61·55-s + 0.794·57-s + 0.260·59-s − 1.02·61-s + 2.48·65-s + 1.71·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8062754636\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8062754636\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 5 | \( 1 + 4 T + p T^{2} \) | 1.5.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 8 T + p T^{2} \) | 1.41.i |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 15 T + p T^{2} \) | 1.79.ap |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 - 18 T + p T^{2} \) | 1.97.as |
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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
−8.412207524319629505390518493394, −7.83914240649366228838486958135, −7.49313014608019165765050404536, −6.64060448648621865259785379284, −5.29448629237218268670726784084, −4.81108611409071186497772433095, −3.76859904237221605053167419340, −3.10298042904571192545973628388, −2.36237586664771651599690165474, −0.48857560730778819962298282627,
0.48857560730778819962298282627, 2.36237586664771651599690165474, 3.10298042904571192545973628388, 3.76859904237221605053167419340, 4.81108611409071186497772433095, 5.29448629237218268670726784084, 6.64060448648621865259785379284, 7.49313014608019165765050404536, 7.83914240649366228838486958135, 8.412207524319629505390518493394
