| L(s) = 1 | − 3·7-s − 3·9-s + 5·11-s − 13-s − 4·17-s + 19-s + 23-s + 5·29-s + 2·31-s + 8·37-s + 3·41-s − 3·43-s − 6·47-s + 2·49-s + 6·53-s + 8·61-s + 9·63-s + 16·67-s − 10·71-s − 3·73-s − 15·77-s + 79-s + 9·81-s + 83-s − 14·89-s + 3·91-s − 10·97-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 9-s + 1.50·11-s − 0.277·13-s − 0.970·17-s + 0.229·19-s + 0.208·23-s + 0.928·29-s + 0.359·31-s + 1.31·37-s + 0.468·41-s − 0.457·43-s − 0.875·47-s + 2/7·49-s + 0.824·53-s + 1.02·61-s + 1.13·63-s + 1.95·67-s − 1.18·71-s − 0.351·73-s − 1.70·77-s + 0.112·79-s + 81-s + 0.109·83-s − 1.48·89-s + 0.314·91-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.472812640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.472812640\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 3 T + p T^{2} \) | 1.43.d |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 - 16 T + p T^{2} \) | 1.67.aq |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + 14 T + p T^{2} \) | 1.89.o |
| 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
−14.67488778350969, −14.50198554499393, −13.85234697184540, −13.28664356128759, −12.89220869808186, −12.14332427721978, −11.78664756259741, −11.27172020720272, −10.77546257546947, −9.844698639783684, −9.637732777877601, −9.050853299365328, −8.520592811094499, −8.038999693859848, −7.001305575663438, −6.710332303604801, −6.229542294131085, −5.667791699369546, −4.861782916795995, −4.156931963066292, −3.627620987445085, −2.851866248987849, −2.437215648917338, −1.317680239519850, −0.4704033404223698,
0.4704033404223698, 1.317680239519850, 2.437215648917338, 2.851866248987849, 3.627620987445085, 4.156931963066292, 4.861782916795995, 5.667791699369546, 6.229542294131085, 6.710332303604801, 7.001305575663438, 8.038999693859848, 8.520592811094499, 9.050853299365328, 9.637732777877601, 9.844698639783684, 10.77546257546947, 11.27172020720272, 11.78664756259741, 12.14332427721978, 12.89220869808186, 13.28664356128759, 13.85234697184540, 14.50198554499393, 14.67488778350969
