| L(s) = 1 | − 4·7-s − 6·11-s − 13-s + 6·17-s + 4·19-s − 3·23-s + 3·29-s + 4·31-s − 2·37-s − 6·41-s − 7·43-s + 9·49-s − 9·53-s − 6·59-s − 61-s + 14·67-s − 6·71-s + 4·73-s + 24·77-s − 11·79-s + 6·83-s + 4·91-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.80·11-s − 0.277·13-s + 1.45·17-s + 0.917·19-s − 0.625·23-s + 0.557·29-s + 0.718·31-s − 0.328·37-s − 0.937·41-s − 1.06·43-s + 9/7·49-s − 1.23·53-s − 0.781·59-s − 0.128·61-s + 1.71·67-s − 0.712·71-s + 0.468·73-s + 2.73·77-s − 1.23·79-s + 0.658·83-s + 0.419·91-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 3 T + p T^{2} \) | 1.23.d |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 11 T + p T^{2} \) | 1.79.l |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−14.94383714127547, −14.14201369228056, −13.85361975425680, −13.23776764671028, −12.80830019316348, −12.38728094504100, −11.88614241788527, −11.30240468992635, −10.32930477571877, −10.16197108398616, −9.888643965776918, −9.225617982956614, −8.471173933199031, −7.815643912774406, −7.616849108512087, −6.787321282663176, −6.306069851704356, −5.623253013934475, −5.198526825782010, −4.607641188021187, −3.518482932132103, −3.185674535941393, −2.743591114664634, −1.845945081518659, −0.7545717209561546, 0,
0.7545717209561546, 1.845945081518659, 2.743591114664634, 3.185674535941393, 3.518482932132103, 4.607641188021187, 5.198526825782010, 5.623253013934475, 6.306069851704356, 6.787321282663176, 7.616849108512087, 7.815643912774406, 8.471173933199031, 9.225617982956614, 9.888643965776918, 10.16197108398616, 10.32930477571877, 11.30240468992635, 11.88614241788527, 12.38728094504100, 12.80830019316348, 13.23776764671028, 13.85361975425680, 14.14201369228056, 14.94383714127547
