| L(s) = 1 | − 3-s − 7-s + 9-s − 6·11-s + 5·13-s + 6·17-s − 5·19-s + 21-s − 6·23-s − 27-s + 31-s + 6·33-s − 2·37-s − 5·39-s + 43-s − 6·47-s − 6·49-s − 6·51-s − 12·53-s + 5·57-s − 6·59-s + 13·61-s − 63-s + 11·67-s + 6·69-s − 2·73-s + 6·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.80·11-s + 1.38·13-s + 1.45·17-s − 1.14·19-s + 0.218·21-s − 1.25·23-s − 0.192·27-s + 0.179·31-s + 1.04·33-s − 0.328·37-s − 0.800·39-s + 0.152·43-s − 0.875·47-s − 6/7·49-s − 0.840·51-s − 1.64·53-s + 0.662·57-s − 0.781·59-s + 1.66·61-s − 0.125·63-s + 1.34·67-s + 0.722·69-s − 0.234·73-s + 0.683·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 31 | \( 1 - T + p T^{2} \) | 1.31.ab |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 13 T + p T^{2} \) | 1.61.an |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−12.96919586324453, −12.72273414328974, −12.25547093982971, −11.60339762673720, −11.21373009844449, −10.69242762776121, −10.39837207846271, −9.858303940877207, −9.658016216426545, −8.740835162615127, −8.211275659562668, −8.055488247929635, −7.527336186922696, −6.808432225863676, −6.336020994930463, −5.865533746973617, −5.573575618807301, −4.940471918077882, −4.471857220601681, −3.720336057719464, −3.359096434112124, −2.746643532235687, −2.010657118144172, −1.481864199719130, −0.6198334290614709, 0,
0.6198334290614709, 1.481864199719130, 2.010657118144172, 2.746643532235687, 3.359096434112124, 3.720336057719464, 4.471857220601681, 4.940471918077882, 5.573575618807301, 5.865533746973617, 6.336020994930463, 6.808432225863676, 7.527336186922696, 8.055488247929635, 8.211275659562668, 8.740835162615127, 9.658016216426545, 9.858303940877207, 10.39837207846271, 10.69242762776121, 11.21373009844449, 11.60339762673720, 12.25547093982971, 12.72273414328974, 12.96919586324453
