| L(s) = 1 | − 2-s − 3-s + 4-s + 5-s + 6-s − 8-s + 9-s − 10-s − 12-s − 13-s − 15-s + 16-s + 3·17-s − 18-s + 19-s + 20-s + 24-s + 25-s + 26-s − 27-s + 6·29-s + 30-s − 2·31-s − 32-s − 3·34-s + 36-s − 10·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.277·13-s − 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.204·24-s + 1/5·25-s + 0.196·26-s − 0.192·27-s + 1.11·29-s + 0.182·30-s − 0.359·31-s − 0.176·32-s − 0.514·34-s + 1/6·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 + 15 T + p T^{2} \) | 1.83.p |
| 89 | \( 1 - 9 T + p T^{2} \) | 1.89.aj |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−16.14115084861133, −15.48464161543470, −15.17784871163792, −14.21046029972372, −13.96805590749418, −13.21821994631914, −12.47238342252438, −12.04016281622712, −11.69187856730811, −10.72576455461779, −10.42528595551166, −10.03389630925974, −9.172192605458474, −8.894071240684458, −8.032844759019227, −7.440111903325041, −6.907070237819548, −6.253793739692680, −5.628373714348232, −5.106178734058118, −4.305603305060868, −3.372164179171839, −2.668544552420986, −1.738202952058800, −1.061183527602555, 0,
1.061183527602555, 1.738202952058800, 2.668544552420986, 3.372164179171839, 4.305603305060868, 5.106178734058118, 5.628373714348232, 6.253793739692680, 6.907070237819548, 7.440111903325041, 8.032844759019227, 8.894071240684458, 9.172192605458474, 10.03389630925974, 10.42528595551166, 10.72576455461779, 11.69187856730811, 12.04016281622712, 12.47238342252438, 13.21821994631914, 13.96805590749418, 14.21046029972372, 15.17784871163792, 15.48464161543470, 16.14115084861133
