| L(s) = 1 | + 5·7-s − 7·13-s − 19-s − 5·25-s − 4·31-s − 37-s + 8·43-s + 18·49-s − 13·61-s + 11·67-s + 17·73-s − 13·79-s − 35·91-s + 5·97-s − 7·103-s + 2·109-s + ⋯ |
| L(s) = 1 | + 1.88·7-s − 1.94·13-s − 0.229·19-s − 25-s − 0.718·31-s − 0.164·37-s + 1.21·43-s + 18/7·49-s − 1.66·61-s + 1.34·67-s + 1.98·73-s − 1.46·79-s − 3.66·91-s + 0.507·97-s − 0.689·103-s + 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.112912674\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.112912674\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 5 T + p T^{2} \) | 1.7.af |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 7 T + p T^{2} \) | 1.13.h |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 17 T + p T^{2} \) | 1.73.ar |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−14.00533773800539081851320941315, −12.48081907715316673289028111467, −11.64280365407061309217144113778, −10.67338269741490887569061453351, −9.420824192029041109275264269235, −8.079851394884847745055740317792, −7.31346310196346190295346056081, −5.41619595580815395232018816358, −4.44194996130619826314058940000, −2.12496210928289498867796346468,
2.12496210928289498867796346468, 4.44194996130619826314058940000, 5.41619595580815395232018816358, 7.31346310196346190295346056081, 8.079851394884847745055740317792, 9.420824192029041109275264269235, 10.67338269741490887569061453351, 11.64280365407061309217144113778, 12.48081907715316673289028111467, 14.00533773800539081851320941315
