| L(s) = 1 | + 7-s − 6·11-s − 6·13-s − 2·17-s − 7·19-s + 8·23-s + 6·29-s + 9·31-s − 3·37-s − 10·41-s − 43-s − 2·47-s − 6·49-s − 2·53-s − 12·59-s + 3·61-s − 4·67-s + 12·71-s + 11·73-s − 6·77-s + 11·79-s − 6·83-s − 8·89-s − 6·91-s + 7·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.80·11-s − 1.66·13-s − 0.485·17-s − 1.60·19-s + 1.66·23-s + 1.11·29-s + 1.61·31-s − 0.493·37-s − 1.56·41-s − 0.152·43-s − 0.291·47-s − 6/7·49-s − 0.274·53-s − 1.56·59-s + 0.384·61-s − 0.488·67-s + 1.42·71-s + 1.28·73-s − 0.683·77-s + 1.23·79-s − 0.658·83-s − 0.847·89-s − 0.628·91-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.018023890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018023890\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 - T + p T^{2} \) | 1.7.ab |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−16.70183651477926, −15.76086953753945, −15.25091677492862, −15.08585677658781, −14.26800539358931, −13.58760532769052, −13.09688080260559, −12.46450435217019, −12.11110582574428, −11.12519731827859, −10.73000393157755, −10.14406940960383, −9.645860034658084, −8.668649357299778, −8.219725930928004, −7.697085500327796, −6.828761137734445, −6.474150240541850, −5.236847954350409, −4.907350251067635, −4.478823175189329, −3.088388791320817, −2.616113581122247, −1.884739934608634, −0.4414512663399209,
0.4414512663399209, 1.884739934608634, 2.616113581122247, 3.088388791320817, 4.478823175189329, 4.907350251067635, 5.236847954350409, 6.474150240541850, 6.828761137734445, 7.697085500327796, 8.219725930928004, 8.668649357299778, 9.645860034658084, 10.14406940960383, 10.73000393157755, 11.12519731827859, 12.11110582574428, 12.46450435217019, 13.09688080260559, 13.58760532769052, 14.26800539358931, 15.08585677658781, 15.25091677492862, 15.76086953753945, 16.70183651477926
