| L(s) = 1 | − 2·3-s − 5-s + 4·7-s + 9-s − 2·13-s + 2·15-s − 4·17-s − 8·21-s + 25-s + 4·27-s − 29-s − 4·35-s − 4·37-s + 4·39-s − 2·41-s + 2·43-s − 45-s + 10·47-s + 9·49-s + 8·51-s − 6·53-s − 4·59-s − 6·61-s + 4·63-s + 2·65-s − 8·71-s + 8·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 0.554·13-s + 0.516·15-s − 0.970·17-s − 1.74·21-s + 1/5·25-s + 0.769·27-s − 0.185·29-s − 0.676·35-s − 0.657·37-s + 0.640·39-s − 0.312·41-s + 0.304·43-s − 0.149·45-s + 1.45·47-s + 9/7·49-s + 1.12·51-s − 0.824·53-s − 0.520·59-s − 0.768·61-s + 0.503·63-s + 0.248·65-s − 0.949·71-s + 0.936·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 29 | \( 1 + T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 10 T + p T^{2} \) | 1.47.ak |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 97 | \( 1 - 4 T + p T^{2} \) | 1.97.ae |
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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
−8.537274146959938627933686681301, −7.78389070447038109806443436473, −7.07106551392112613337824673886, −6.20663438914401278080854834579, −5.27910238999248137212871110287, −4.79856778545261567868684284810, −4.03884898650944165711652169657, −2.55383475490914048408219792962, −1.38105238780744038949127273447, 0,
1.38105238780744038949127273447, 2.55383475490914048408219792962, 4.03884898650944165711652169657, 4.79856778545261567868684284810, 5.27910238999248137212871110287, 6.20663438914401278080854834579, 7.07106551392112613337824673886, 7.78389070447038109806443436473, 8.537274146959938627933686681301
