| L(s) = 1 | + 3·11-s + 4·13-s + 3·17-s + 4·19-s − 2·23-s + 2·29-s − 4·31-s − 4·37-s + 10·41-s − 7·43-s + 4·47-s − 7·49-s − 8·53-s − 59-s + 4·61-s + 12·67-s + 14·71-s + 2·73-s + 83-s − 89-s − 17·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 0.904·11-s + 1.10·13-s + 0.727·17-s + 0.917·19-s − 0.417·23-s + 0.371·29-s − 0.718·31-s − 0.657·37-s + 1.56·41-s − 1.06·43-s + 0.583·47-s − 49-s − 1.09·53-s − 0.130·59-s + 0.512·61-s + 1.46·67-s + 1.66·71-s + 0.234·73-s + 0.109·83-s − 0.105·89-s − 1.72·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.182697610\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.182697610\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 8 T + p T^{2} \) | 1.53.i |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 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
−14.17717753935991, −13.88254713921592, −13.27200797787669, −12.55687977077916, −12.35038190480274, −11.62916275097669, −11.18645848563690, −10.86642482066077, −10.00773720975272, −9.670961912143463, −9.146550735658061, −8.588431203009664, −7.996749655436801, −7.620461793315970, −6.693641417139246, −6.569152852248201, −5.711311460652997, −5.382151915798350, −4.606154906991823, −3.840968297926015, −3.545212794154723, −2.874033424023561, −1.908802572784128, −1.316600211765433, −0.6525794578142189,
0.6525794578142189, 1.316600211765433, 1.908802572784128, 2.874033424023561, 3.545212794154723, 3.840968297926015, 4.606154906991823, 5.382151915798350, 5.711311460652997, 6.569152852248201, 6.693641417139246, 7.620461793315970, 7.996749655436801, 8.588431203009664, 9.146550735658061, 9.670961912143463, 10.00773720975272, 10.86642482066077, 11.18645848563690, 11.62916275097669, 12.35038190480274, 12.55687977077916, 13.27200797787669, 13.88254713921592, 14.17717753935991
