| L(s) = 1 | + 3-s − 2·7-s + 9-s + 11-s − 4·13-s − 17-s − 8·19-s − 2·21-s − 6·23-s − 5·25-s + 27-s + 2·29-s − 4·31-s + 33-s + 6·37-s − 4·39-s + 10·41-s − 4·43-s + 2·47-s − 3·49-s − 51-s − 12·53-s − 8·57-s − 12·61-s − 2·63-s + 12·67-s − 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.242·17-s − 1.83·19-s − 0.436·21-s − 1.25·23-s − 25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s + 0.174·33-s + 0.986·37-s − 0.640·39-s + 1.56·41-s − 0.609·43-s + 0.291·47-s − 3/7·49-s − 0.140·51-s − 1.64·53-s − 1.05·57-s − 1.53·61-s − 0.251·63-s + 1.46·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8999284745\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8999284745\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 3 | \( 1 - T \) | |
| 11 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.89019960745929, −14.31290714082034, −14.07414246964492, −13.21451284658241, −12.88313432052339, −12.37977086353777, −11.95339765541844, −11.10873748491390, −10.67576688558010, −9.962680195829979, −9.498449055451165, −9.265192690524721, −8.338646089558109, −7.983014203553383, −7.403286420627668, −6.632533535779711, −6.270844306932208, −5.671674824555361, −4.667922565833749, −4.232739627893855, −3.680976361544099, −2.825341285591685, −2.264688580307339, −1.682831446379166, −0.3166184168518901,
0.3166184168518901, 1.682831446379166, 2.264688580307339, 2.825341285591685, 3.680976361544099, 4.232739627893855, 4.667922565833749, 5.671674824555361, 6.270844306932208, 6.632533535779711, 7.403286420627668, 7.983014203553383, 8.338646089558109, 9.265192690524721, 9.498449055451165, 9.962680195829979, 10.67576688558010, 11.10873748491390, 11.95339765541844, 12.37977086353777, 12.88313432052339, 13.21451284658241, 14.07414246964492, 14.31290714082034, 14.89019960745929
