| L(s) = 1 | − 2·5-s − 4·7-s + 4·11-s − 2·17-s − 8·19-s − 25-s + 6·29-s + 4·31-s + 8·35-s − 2·37-s − 10·41-s − 4·43-s + 8·47-s + 9·49-s − 10·53-s − 8·55-s − 4·59-s + 2·61-s − 16·67-s − 8·71-s − 2·73-s − 16·77-s + 8·79-s − 12·83-s + 4·85-s + 14·89-s + 16·95-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.51·7-s + 1.20·11-s − 0.485·17-s − 1.83·19-s − 1/5·25-s + 1.11·29-s + 0.718·31-s + 1.35·35-s − 0.328·37-s − 1.56·41-s − 0.609·43-s + 1.16·47-s + 9/7·49-s − 1.37·53-s − 1.07·55-s − 0.520·59-s + 0.256·61-s − 1.95·67-s − 0.949·71-s − 0.234·73-s − 1.82·77-s + 0.900·79-s − 1.31·83-s + 0.433·85-s + 1.48·89-s + 1.64·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1801824832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1801824832\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 16 T + p T^{2} \) | 1.67.q |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.63729508908547, −13.36440537054196, −12.65901533212323, −12.32395443077363, −11.91659038240500, −11.44956947978151, −10.81950519070993, −10.18200372616393, −10.02110359879624, −9.175161209512654, −8.784631637318892, −8.476524240325583, −7.732616645660084, −7.102189623721945, −6.579798706213296, −6.329903948653145, −5.885866133102760, −4.699951360135816, −4.479925747343163, −3.774545152990904, −3.408003265780723, −2.773355988334470, −2.008828591072030, −1.176608224139180, −0.1414147712047156,
0.1414147712047156, 1.176608224139180, 2.008828591072030, 2.773355988334470, 3.408003265780723, 3.774545152990904, 4.479925747343163, 4.699951360135816, 5.885866133102760, 6.329903948653145, 6.579798706213296, 7.102189623721945, 7.732616645660084, 8.476524240325583, 8.784631637318892, 9.175161209512654, 10.02110359879624, 10.18200372616393, 10.81950519070993, 11.44956947978151, 11.91659038240500, 12.32395443077363, 12.65901533212323, 13.36440537054196, 13.63729508908547
