| L(s) = 1 | + 3-s − 2·5-s + 9-s − 6·13-s − 2·15-s − 2·17-s − 8·19-s − 4·23-s − 25-s + 27-s − 2·29-s + 8·31-s + 6·37-s − 6·39-s − 2·41-s − 8·43-s − 2·45-s + 4·47-s − 2·51-s + 2·53-s − 8·57-s + 12·59-s + 10·61-s + 12·65-s − 12·67-s − 4·69-s − 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s − 1.66·13-s − 0.516·15-s − 0.485·17-s − 1.83·19-s − 0.834·23-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.986·37-s − 0.960·39-s − 0.312·41-s − 1.21·43-s − 0.298·45-s + 0.583·47-s − 0.280·51-s + 0.274·53-s − 1.05·57-s + 1.56·59-s + 1.28·61-s + 1.48·65-s − 1.46·67-s − 0.481·69-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142296 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−13.57431589138854, −13.15593300170273, −12.68843069977389, −12.12930792227636, −11.81874264590076, −11.38246536899960, −10.68834332473791, −10.09551246970778, −9.957318721414741, −9.235436561300314, −8.647604520987292, −8.289372179210355, −7.787955143599875, −7.439789028256273, −6.716383895243574, −6.458765746305795, −5.675120314898625, −4.912122621904276, −4.480579745380510, −4.061742161610010, −3.565313015387512, −2.661886532121407, −2.351970354408068, −1.791707254549638, −0.6351970302252066, 0,
0.6351970302252066, 1.791707254549638, 2.351970354408068, 2.661886532121407, 3.565313015387512, 4.061742161610010, 4.480579745380510, 4.912122621904276, 5.675120314898625, 6.458765746305795, 6.716383895243574, 7.439789028256273, 7.787955143599875, 8.289372179210355, 8.647604520987292, 9.235436561300314, 9.957318721414741, 10.09551246970778, 10.68834332473791, 11.38246536899960, 11.81874264590076, 12.12930792227636, 12.68843069977389, 13.15593300170273, 13.57431589138854
