| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s − 7-s + 8-s + 9-s − 12-s + 4·13-s − 14-s + 16-s − 17-s + 18-s + 2·19-s + 21-s − 6·23-s − 24-s + 4·26-s − 27-s − 28-s − 4·31-s + 32-s − 34-s + 36-s + 4·37-s + 2·38-s − 4·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s + 0.218·21-s − 1.25·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s − 0.188·28-s − 0.718·31-s + 0.176·32-s − 0.171·34-s + 1/6·36-s + 0.657·37-s + 0.324·38-s − 0.640·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.709732905\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.709732905\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−15.93967328980765, −15.37140162688512, −14.65747893867627, −14.05853826830567, −13.65271118924176, −12.86424321781939, −12.73236292603794, −11.90393382837549, −11.44588335407632, −10.91541235761321, −10.43153568969608, −9.638990364102186, −9.209244834236118, −8.247764108502827, −7.768891983596329, −7.004788480304444, −6.315510189736657, −5.995176487349081, −5.349182716565686, −4.596625306378115, −3.893732617099481, −3.432142398564531, −2.454910346662638, −1.631591999295443, −0.6485687402843876,
0.6485687402843876, 1.631591999295443, 2.454910346662638, 3.432142398564531, 3.893732617099481, 4.596625306378115, 5.349182716565686, 5.995176487349081, 6.315510189736657, 7.004788480304444, 7.768891983596329, 8.247764108502827, 9.209244834236118, 9.638990364102186, 10.43153568969608, 10.91541235761321, 11.44588335407632, 11.90393382837549, 12.73236292603794, 12.86424321781939, 13.65271118924176, 14.05853826830567, 14.65747893867627, 15.37140162688512, 15.93967328980765
