| L(s) = 1 | + 7-s − 2·13-s + 17-s − 2·29-s + 8·31-s + 6·37-s + 6·41-s − 4·43-s + 49-s + 14·53-s + 8·59-s + 14·61-s − 4·67-s − 8·71-s − 10·73-s + 8·79-s − 16·83-s − 2·89-s − 2·91-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 119-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.554·13-s + 0.242·17-s − 0.371·29-s + 1.43·31-s + 0.986·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 1.92·53-s + 1.04·59-s + 1.79·61-s − 0.488·67-s − 0.949·71-s − 1.17·73-s + 0.900·79-s − 1.75·83-s − 0.211·89-s − 0.209·91-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 0.0916·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 17 | \( 1 - T \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−13.14400101829847, −12.93386382198948, −12.12419507181885, −11.83385378324797, −11.50971958733144, −10.88704412789098, −10.40634393725342, −9.875176659843537, −9.662520900450840, −8.903259114597721, −8.506959485216135, −8.035939465066578, −7.542593288768036, −7.025701492882430, −6.627841262955354, −5.877085019718140, −5.537429828181852, −4.993080037269705, −4.305773821942194, −4.078257628072422, −3.286721236383161, −2.512445845409600, −2.390089685026802, −1.358696622905816, −0.9049307646946152, 0,
0.9049307646946152, 1.358696622905816, 2.390089685026802, 2.512445845409600, 3.286721236383161, 4.078257628072422, 4.305773821942194, 4.993080037269705, 5.537429828181852, 5.877085019718140, 6.627841262955354, 7.025701492882430, 7.542593288768036, 8.035939465066578, 8.506959485216135, 8.903259114597721, 9.662520900450840, 9.875176659843537, 10.40634393725342, 10.88704412789098, 11.50971958733144, 11.83385378324797, 12.12419507181885, 12.93386382198948, 13.14400101829847
