| L(s) = 1 | − 6·11-s + 2·13-s − 17-s + 8·19-s − 4·23-s + 6·29-s + 10·31-s − 6·37-s + 10·41-s − 2·43-s − 6·47-s − 7·49-s + 10·53-s − 2·61-s − 10·67-s − 14·71-s + 6·73-s + 10·79-s + 14·83-s + 10·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.80·11-s + 0.554·13-s − 0.242·17-s + 1.83·19-s − 0.834·23-s + 1.11·29-s + 1.79·31-s − 0.986·37-s + 1.56·41-s − 0.304·43-s − 0.875·47-s − 49-s + 1.37·53-s − 0.256·61-s − 1.22·67-s − 1.66·71-s + 0.702·73-s + 1.12·79-s + 1.53·83-s + 1.05·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.396689446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.396689446\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 2 T + p T^{2} \) | 1.61.c |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 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
−13.02775695870746, −12.32488355846916, −11.85333246616633, −11.69102719750611, −10.90201296859289, −10.50060761654204, −10.16115682547456, −9.737540663492299, −9.161412209039272, −8.553119308274101, −8.120180448925674, −7.733667954202136, −7.370393371750062, −6.654371887064986, −6.127565989561395, −5.734626911618013, −4.963488305254445, −4.912190042772617, −4.159898907097029, −3.384356625546494, −2.991954472996044, −2.519653055560614, −1.837883796275881, −1.023914487509259, −0.4810421280698036,
0.4810421280698036, 1.023914487509259, 1.837883796275881, 2.519653055560614, 2.991954472996044, 3.384356625546494, 4.159898907097029, 4.912190042772617, 4.963488305254445, 5.734626911618013, 6.127565989561395, 6.654371887064986, 7.370393371750062, 7.733667954202136, 8.120180448925674, 8.553119308274101, 9.161412209039272, 9.737540663492299, 10.16115682547456, 10.50060761654204, 10.90201296859289, 11.69102719750611, 11.85333246616633, 12.32488355846916, 13.02775695870746
