| L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s − 5·13-s − 14-s + 16-s + 3·17-s + 2·19-s + 20-s + 7·23-s − 4·25-s − 5·26-s − 28-s − 6·29-s + 32-s + 3·34-s − 35-s − 10·37-s + 2·38-s + 40-s + 5·41-s + 2·43-s + 7·46-s − 8·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s − 1.38·13-s − 0.267·14-s + 1/4·16-s + 0.727·17-s + 0.458·19-s + 0.223·20-s + 1.45·23-s − 4/5·25-s − 0.980·26-s − 0.188·28-s − 1.11·29-s + 0.176·32-s + 0.514·34-s − 0.169·35-s − 1.64·37-s + 0.324·38-s + 0.158·40-s + 0.780·41-s + 0.304·43-s + 1.03·46-s − 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.345508961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.345508961\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 - T + p T^{2} \) | 1.5.ab |
| 13 | \( 1 + 5 T + p T^{2} \) | 1.13.f |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 - 7 T + p T^{2} \) | 1.23.ah |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 5 T + p T^{2} \) | 1.41.af |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 15 T + p T^{2} \) | 1.67.ap |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 7 T + p T^{2} \) | 1.79.ah |
| 83 | \( 1 + 3 T + p T^{2} \) | 1.83.d |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−14.55132884182597, −14.08768371248801, −13.66820127693222, −13.05008284605269, −12.52706610442022, −12.30531694263497, −11.56500285784724, −11.11908115986258, −10.46038135129383, −9.914248028595537, −9.456701853461190, −9.057408277400841, −8.111733881639202, −7.525072191142771, −7.148525356071300, −6.542225588345487, −5.886247669704801, −5.204367162241018, −5.071106192278298, −4.182331603526941, −3.407500509027451, −3.022804045361273, −2.208162697076957, −1.627224518264375, −0.5575921817199312,
0.5575921817199312, 1.627224518264375, 2.208162697076957, 3.022804045361273, 3.407500509027451, 4.182331603526941, 5.071106192278298, 5.204367162241018, 5.886247669704801, 6.542225588345487, 7.148525356071300, 7.525072191142771, 8.111733881639202, 9.057408277400841, 9.456701853461190, 9.914248028595537, 10.46038135129383, 11.11908115986258, 11.56500285784724, 12.30531694263497, 12.52706610442022, 13.05008284605269, 13.66820127693222, 14.08768371248801, 14.55132884182597
