| L(s) = 1 | + 2·5-s − 2·13-s + 2·17-s − 25-s + 4·31-s − 2·37-s + 6·41-s − 4·43-s − 4·47-s − 7·49-s + 6·53-s + 2·59-s + 2·61-s − 4·65-s + 4·67-s + 2·71-s − 10·73-s − 4·83-s + 4·85-s + 6·89-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.554·13-s + 0.485·17-s − 1/5·25-s + 0.718·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.583·47-s − 49-s + 0.824·53-s + 0.260·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.237·71-s − 1.17·73-s − 0.439·83-s + 0.433·85-s + 0.635·89-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.775499667\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.775499667\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 397 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 2 T + p T^{2} \) | 1.71.ac |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−13.05066461480492, −12.49454894313997, −12.03893653163600, −11.51262454751886, −11.18360404853898, −10.39766062854085, −10.06029585288417, −9.839696303016513, −9.207193343670080, −8.822294231536322, −8.180480509905471, −7.763874103038967, −7.234855589630521, −6.640632689400990, −6.260392344310874, −5.657809538782513, −5.312949250248774, −4.709045514869224, −4.225459089260334, −3.476454603754469, −2.991189918053028, −2.331615520220836, −1.880311802454469, −1.214177546690783, −0.4659257555723944,
0.4659257555723944, 1.214177546690783, 1.880311802454469, 2.331615520220836, 2.991189918053028, 3.476454603754469, 4.225459089260334, 4.709045514869224, 5.312949250248774, 5.657809538782513, 6.260392344310874, 6.640632689400990, 7.234855589630521, 7.763874103038967, 8.180480509905471, 8.822294231536322, 9.207193343670080, 9.839696303016513, 10.06029585288417, 10.39766062854085, 11.18360404853898, 11.51262454751886, 12.03893653163600, 12.49454894313997, 13.05066461480492
