| L(s) = 1 | − 3-s − 5-s − 7-s + 9-s − 4·13-s + 15-s + 5·17-s − 6·19-s + 21-s − 4·25-s − 27-s − 10·29-s + 2·31-s + 35-s − 2·37-s + 4·39-s − 2·41-s + 13·43-s − 45-s − 47-s + 49-s − 5·51-s + 4·53-s + 6·57-s + 3·59-s + 8·61-s − 63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s − 1.10·13-s + 0.258·15-s + 1.21·17-s − 1.37·19-s + 0.218·21-s − 4/5·25-s − 0.192·27-s − 1.85·29-s + 0.359·31-s + 0.169·35-s − 0.328·37-s + 0.640·39-s − 0.312·41-s + 1.98·43-s − 0.149·45-s − 0.145·47-s + 1/7·49-s − 0.700·51-s + 0.549·53-s + 0.794·57-s + 0.390·59-s + 1.02·61-s − 0.125·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40656 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 5 T + p T^{2} \) | 1.17.af |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 10 T + p T^{2} \) | 1.29.k |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 13 T + p T^{2} \) | 1.43.an |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 - T + p T^{2} \) | 1.89.ab |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−15.06934870034301, −14.55546031019208, −14.12196058950221, −13.29951081611920, −12.79316552236926, −12.45969362122783, −11.90439918823390, −11.46293745656863, −10.84505368853832, −10.30799189823661, −9.807215473019103, −9.355043070480404, −8.640769618692807, −7.955149502830108, −7.431805370977795, −7.076174136657233, −6.283259868040659, −5.712423378016332, −5.301399568927114, −4.462671109663828, −3.967424941291893, −3.377536074154924, −2.440193718292206, −1.869809910953147, −0.7413004324795949, 0,
0.7413004324795949, 1.869809910953147, 2.440193718292206, 3.377536074154924, 3.967424941291893, 4.462671109663828, 5.301399568927114, 5.712423378016332, 6.283259868040659, 7.076174136657233, 7.431805370977795, 7.955149502830108, 8.640769618692807, 9.355043070480404, 9.807215473019103, 10.30799189823661, 10.84505368853832, 11.46293745656863, 11.90439918823390, 12.45969362122783, 12.79316552236926, 13.29951081611920, 14.12196058950221, 14.55546031019208, 15.06934870034301
