| L(s) = 1 | − 3·3-s + 3·5-s + 3·7-s + 6·9-s + 11-s − 9·15-s − 5·17-s + 4·19-s − 9·21-s + 8·23-s + 4·25-s − 9·27-s − 3·33-s + 9·35-s − 5·37-s − 11·43-s + 18·45-s + 9·47-s + 2·49-s + 15·51-s + 6·53-s + 3·55-s − 12·57-s + 6·59-s + 18·63-s + 8·67-s − 24·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 1.34·5-s + 1.13·7-s + 2·9-s + 0.301·11-s − 2.32·15-s − 1.21·17-s + 0.917·19-s − 1.96·21-s + 1.66·23-s + 4/5·25-s − 1.73·27-s − 0.522·33-s + 1.52·35-s − 0.821·37-s − 1.67·43-s + 2.68·45-s + 1.31·47-s + 2/7·49-s + 2.10·51-s + 0.824·53-s + 0.404·55-s − 1.58·57-s + 0.781·59-s + 2.26·63-s + 0.977·67-s − 2.88·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 118976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.642039271\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.642039271\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T + p T^{2} \) | 1.3.d |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 17 | \( 1 + 5 T + p T^{2} \) | 1.17.f |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−13.41653913975316, −13.07971476124055, −12.64502622384377, −11.88913634827472, −11.60149764587050, −11.17316253585804, −10.78543308440226, −10.27253970882044, −9.863083509252523, −9.242655668768880, −8.780112830825999, −8.220903609823897, −7.184455399184019, −7.081735215979893, −6.487499596925948, −5.969498543308537, −5.400696749021594, −5.056449353649197, −4.791502129681341, −4.055356392493421, −3.218743562763490, −2.225919973163891, −1.820421323515311, −1.129658951041164, −0.6308625631267772,
0.6308625631267772, 1.129658951041164, 1.820421323515311, 2.225919973163891, 3.218743562763490, 4.055356392493421, 4.791502129681341, 5.056449353649197, 5.400696749021594, 5.969498543308537, 6.487499596925948, 7.081735215979893, 7.184455399184019, 8.220903609823897, 8.780112830825999, 9.242655668768880, 9.863083509252523, 10.27253970882044, 10.78543308440226, 11.17316253585804, 11.60149764587050, 11.88913634827472, 12.64502622384377, 13.07971476124055, 13.41653913975316
