| L(s) = 1 | − 2-s + 4-s − 4·7-s − 8-s − 3·11-s − 4·13-s + 4·14-s + 16-s + 3·17-s + 7·19-s + 3·22-s + 6·23-s − 5·25-s + 4·26-s − 4·28-s + 4·31-s − 32-s − 3·34-s − 8·37-s − 7·38-s + 3·41-s + 7·43-s − 3·44-s − 6·46-s − 6·47-s + 9·49-s + 5·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.51·7-s − 0.353·8-s − 0.904·11-s − 1.10·13-s + 1.06·14-s + 1/4·16-s + 0.727·17-s + 1.60·19-s + 0.639·22-s + 1.25·23-s − 25-s + 0.784·26-s − 0.755·28-s + 0.718·31-s − 0.176·32-s − 0.514·34-s − 1.31·37-s − 1.13·38-s + 0.468·41-s + 1.06·43-s − 0.452·44-s − 0.884·46-s − 0.875·47-s + 9/7·49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136242 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 29 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 - 13 T + p T^{2} \) | 1.73.an |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 + 3 T + p T^{2} \) | 1.89.d |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.56726217640306, −13.20162567148839, −12.60366457595513, −12.17822599615816, −11.92960146733391, −11.14320244406291, −10.69813746126092, −10.07745438831372, −9.791613378664412, −9.424003160156983, −9.089532555016401, −8.131922587043632, −7.874450277303989, −7.286141070146792, −6.894051689318821, −6.432463724917644, −5.598457120194715, −5.367091365345168, −4.777542624453350, −3.755403441903378, −3.335596751848657, −2.735441224299248, −2.442462627914383, −1.391548451608221, −0.6695809535238732, 0,
0.6695809535238732, 1.391548451608221, 2.442462627914383, 2.735441224299248, 3.335596751848657, 3.755403441903378, 4.777542624453350, 5.367091365345168, 5.598457120194715, 6.432463724917644, 6.894051689318821, 7.286141070146792, 7.874450277303989, 8.131922587043632, 9.089532555016401, 9.424003160156983, 9.791613378664412, 10.07745438831372, 10.69813746126092, 11.14320244406291, 11.92960146733391, 12.17822599615816, 12.60366457595513, 13.20162567148839, 13.56726217640306
