| L(s) = 1 | − 7-s − 2·11-s − 4·13-s − 2·17-s + 23-s + 2·29-s + 4·37-s − 6·41-s − 2·43-s + 4·47-s + 49-s − 2·59-s − 10·61-s − 2·67-s + 8·71-s + 6·73-s + 2·77-s − 10·89-s + 4·91-s − 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 2·119-s + ⋯ |
| L(s) = 1 | − 0.377·7-s − 0.603·11-s − 1.10·13-s − 0.485·17-s + 0.208·23-s + 0.371·29-s + 0.657·37-s − 0.937·41-s − 0.304·43-s + 0.583·47-s + 1/7·49-s − 0.260·59-s − 1.28·61-s − 0.244·67-s + 0.949·71-s + 0.702·73-s + 0.227·77-s − 1.05·89-s + 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.183·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6883808813\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6883808813\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 23 | \( 1 - T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 + 2 T + p T^{2} \) | 1.67.c |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.71945480697620, −12.23671666999455, −11.93168272472856, −11.34180142025787, −10.79366684707569, −10.46612174747913, −9.962240911932842, −9.421656867468152, −9.238931686046432, −8.436813470744763, −8.120509760084397, −7.585008227237914, −7.035694271840004, −6.735781042916337, −6.114779006459366, −5.584983693342946, −5.052237697126763, −4.645967081874759, −4.114795373568970, −3.437972041540806, −2.844679512736529, −2.476543193485951, −1.853270439322891, −1.076793695224338, −0.2328835094991088,
0.2328835094991088, 1.076793695224338, 1.853270439322891, 2.476543193485951, 2.844679512736529, 3.437972041540806, 4.114795373568970, 4.645967081874759, 5.052237697126763, 5.584983693342946, 6.114779006459366, 6.735781042916337, 7.035694271840004, 7.585008227237914, 8.120509760084397, 8.436813470744763, 9.238931686046432, 9.421656867468152, 9.962240911932842, 10.46612174747913, 10.79366684707569, 11.34180142025787, 11.93168272472856, 12.23671666999455, 12.71945480697620
