| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s + 9-s − 5·11-s − 2·12-s − 13-s + 16-s + 4·17-s + 18-s − 5·19-s − 5·22-s − 23-s − 2·24-s − 26-s + 4·27-s − 2·29-s − 4·31-s + 32-s + 10·33-s + 4·34-s + 36-s − 5·38-s + 2·39-s + 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s + 1/3·9-s − 1.50·11-s − 0.577·12-s − 0.277·13-s + 1/4·16-s + 0.970·17-s + 0.235·18-s − 1.14·19-s − 1.06·22-s − 0.208·23-s − 0.408·24-s − 0.196·26-s + 0.769·27-s − 0.371·29-s − 0.718·31-s + 0.176·32-s + 1.74·33-s + 0.685·34-s + 1/6·36-s − 0.811·38-s + 0.320·39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6621209052\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6621209052\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - T + p T^{2} \) | 1.59.ab |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 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
−14.24712645073841, −13.48372142441783, −13.01094216890672, −12.66603304869485, −12.28157931081701, −11.63360481833614, −11.17899548681497, −10.82504281972024, −10.19579969053525, −9.983594274326951, −9.115709064130865, −8.297662991808601, −7.945118484718053, −7.308247588939761, −6.763353972680545, −6.120519876928632, −5.678749158209142, −5.279265902327832, −4.778288129452700, −4.209291910839969, −3.406277050475167, −2.783150704650598, −2.160338050020467, −1.317085408511460, −0.2579101413914593,
0.2579101413914593, 1.317085408511460, 2.160338050020467, 2.783150704650598, 3.406277050475167, 4.209291910839969, 4.778288129452700, 5.279265902327832, 5.678749158209142, 6.120519876928632, 6.763353972680545, 7.308247588939761, 7.945118484718053, 8.297662991808601, 9.115709064130865, 9.983594274326951, 10.19579969053525, 10.82504281972024, 11.17899548681497, 11.63360481833614, 12.28157931081701, 12.66603304869485, 13.01094216890672, 13.48372142441783, 14.24712645073841
