| L(s) = 1 | + 2-s + 4-s − 5-s − 4·7-s + 8-s − 10-s − 4·11-s − 5·13-s − 4·14-s + 16-s − 17-s − 3·19-s − 20-s − 4·22-s + 8·23-s − 4·25-s − 5·26-s − 4·28-s + 7·31-s + 32-s − 34-s + 4·35-s − 4·37-s − 3·38-s − 40-s + 41-s − 6·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s + 0.353·8-s − 0.316·10-s − 1.20·11-s − 1.38·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s − 0.688·19-s − 0.223·20-s − 0.852·22-s + 1.66·23-s − 4/5·25-s − 0.980·26-s − 0.755·28-s + 1.25·31-s + 0.176·32-s − 0.171·34-s + 0.676·35-s − 0.657·37-s − 0.486·38-s − 0.158·40-s + 0.156·41-s − 0.914·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 41 | \( 1 - T \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 14 T + p T^{2} \) | 1.53.o |
| 59 | \( 1 - 7 T + p T^{2} \) | 1.59.ah |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 5 T + p T^{2} \) | 1.67.f |
| 71 | \( 1 - 7 T + p T^{2} \) | 1.71.ah |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 15 T + p T^{2} \) | 1.83.ap |
| 89 | \( 1 + 13 T + p T^{2} \) | 1.89.n |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−10.08266600804786671706131769377, −9.244347738098059429850983472640, −8.019240374239258929238557865380, −7.14649800311416056102761154095, −6.45057299417612122047272522502, −5.33330380381551699100541521377, −4.47420705672619953222926321067, −3.21932604526646441119983264062, −2.52983349963007098615023775562, 0,
2.52983349963007098615023775562, 3.21932604526646441119983264062, 4.47420705672619953222926321067, 5.33330380381551699100541521377, 6.45057299417612122047272522502, 7.14649800311416056102761154095, 8.019240374239258929238557865380, 9.244347738098059429850983472640, 10.08266600804786671706131769377
