| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s − 3·7-s + 8-s + 9-s − 10-s − 11-s − 12-s − 13-s − 3·14-s + 15-s + 16-s + 18-s − 19-s − 20-s + 3·21-s − 22-s − 24-s − 4·25-s − 26-s − 27-s − 3·28-s + 30-s + 8·31-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s − 1.13·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.301·11-s − 0.288·12-s − 0.277·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.235·18-s − 0.229·19-s − 0.223·20-s + 0.654·21-s − 0.213·22-s − 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s − 0.566·28-s + 0.182·30-s + 1.43·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.64059948951819, −14.08638067674689, −13.49361620684903, −13.13645475389056, −12.57519351663250, −12.14438445628940, −11.80732942367429, −11.14506612433239, −10.63459816088603, −10.14856435587096, −9.596059161573524, −9.134091621748525, −8.155273598487700, −7.853064969523793, −7.090558277947195, −6.621578092469293, −6.192921092738883, −5.592510436282419, −5.047206850101693, −4.298374311347306, −3.945158014867945, −3.148674772263918, −2.679969246419114, −1.848712237125397, −0.8031115269203234, 0,
0.8031115269203234, 1.848712237125397, 2.679969246419114, 3.148674772263918, 3.945158014867945, 4.298374311347306, 5.047206850101693, 5.592510436282419, 6.192921092738883, 6.621578092469293, 7.090558277947195, 7.853064969523793, 8.155273598487700, 9.134091621748525, 9.596059161573524, 10.14856435587096, 10.63459816088603, 11.14506612433239, 11.80732942367429, 12.14438445628940, 12.57519351663250, 13.13645475389056, 13.49361620684903, 14.08638067674689, 14.64059948951819
