| L(s) = 1 | + 7-s + 11-s + 13-s − 3·17-s − 8·19-s − 4·23-s − 3·29-s + 6·31-s + 8·37-s − 10·41-s − 12·43-s − 3·47-s + 49-s + 12·53-s + 2·61-s + 4·67-s − 12·71-s + 10·73-s + 77-s + 13·79-s + 6·89-s + 91-s − 5·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.301·11-s + 0.277·13-s − 0.727·17-s − 1.83·19-s − 0.834·23-s − 0.557·29-s + 1.07·31-s + 1.31·37-s − 1.56·41-s − 1.82·43-s − 0.437·47-s + 1/7·49-s + 1.64·53-s + 0.256·61-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.113·77-s + 1.46·79-s + 0.635·89-s + 0.104·91-s − 0.507·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.534955040\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.534955040\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| good | 11 | \( 1 - T + p T^{2} \) | 1.11.ab |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 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.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 5 T + p T^{2} \) | 1.97.f |
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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
−14.71520742779904, −13.89463484685951, −13.40294395110260, −13.18293091017019, −12.41886455146240, −11.88115771598614, −11.48339372312562, −10.92018946829130, −10.36678747710138, −9.950978160846085, −9.267396234523974, −8.608170707195786, −8.300770426263709, −7.840583777920307, −6.893433654267354, −6.538895793520255, −6.103453870595719, −5.280124665032415, −4.710442667774926, −4.089920371638306, −3.689842229649029, −2.688455859482583, −2.089654155675432, −1.503285173055344, −0.4240837272639244,
0.4240837272639244, 1.503285173055344, 2.089654155675432, 2.688455859482583, 3.689842229649029, 4.089920371638306, 4.710442667774926, 5.280124665032415, 6.103453870595719, 6.538895793520255, 6.893433654267354, 7.840583777920307, 8.300770426263709, 8.608170707195786, 9.267396234523974, 9.950978160846085, 10.36678747710138, 10.92018946829130, 11.48339372312562, 11.88115771598614, 12.41886455146240, 13.18293091017019, 13.40294395110260, 13.89463484685951, 14.71520742779904
