| L(s) = 1 | − 3-s + 9-s + 5·13-s + 6·17-s − 7·19-s − 6·23-s − 27-s − 8·31-s − 37-s − 5·39-s − 8·43-s + 6·47-s − 6·51-s − 6·53-s + 7·57-s − 6·59-s + 61-s + 13·67-s + 6·69-s − 12·71-s − 5·73-s + 7·79-s + 81-s + 18·83-s − 6·89-s + 8·93-s + 7·97-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s + 1.38·13-s + 1.45·17-s − 1.60·19-s − 1.25·23-s − 0.192·27-s − 1.43·31-s − 0.164·37-s − 0.800·39-s − 1.21·43-s + 0.875·47-s − 0.840·51-s − 0.824·53-s + 0.927·57-s − 0.781·59-s + 0.128·61-s + 1.58·67-s + 0.722·69-s − 1.42·71-s − 0.585·73-s + 0.787·79-s + 1/9·81-s + 1.97·83-s − 0.635·89-s + 0.829·93-s + 0.710·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.406023814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.406023814\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - T + p T^{2} \) | 1.61.ab |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 5 T + p T^{2} \) | 1.73.f |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.86279106616597, −12.39287725908031, −12.04721610914052, −11.50818626487308, −10.96475957766670, −10.67809164403328, −10.22520225466460, −9.750106293061544, −9.163322984159538, −8.650018298115504, −8.173745104853740, −7.799173581242862, −7.180679589802778, −6.597081402282082, −6.116541995638510, −5.814895075181796, −5.326990372382020, −4.624839336286340, −4.128019394161459, −3.522742078608354, −3.294529744089385, −2.173806647135101, −1.795964134235755, −1.136837633656596, −0.3604775758290259,
0.3604775758290259, 1.136837633656596, 1.795964134235755, 2.173806647135101, 3.294529744089385, 3.522742078608354, 4.128019394161459, 4.624839336286340, 5.326990372382020, 5.814895075181796, 6.116541995638510, 6.597081402282082, 7.180679589802778, 7.799173581242862, 8.173745104853740, 8.650018298115504, 9.163322984159538, 9.750106293061544, 10.22520225466460, 10.67809164403328, 10.96475957766670, 11.50818626487308, 12.04721610914052, 12.39287725908031, 12.86279106616597
