| L(s) = 1 | + 3·3-s + 6·9-s − 5·11-s − 3·13-s − 17-s − 6·19-s − 6·23-s + 9·27-s − 9·29-s + 4·31-s − 15·33-s − 2·37-s − 9·39-s + 4·41-s − 10·43-s − 47-s − 3·51-s − 4·53-s − 18·57-s + 8·59-s + 8·61-s − 12·67-s − 18·69-s + 8·71-s + 2·73-s + 13·79-s + 9·81-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 2·9-s − 1.50·11-s − 0.832·13-s − 0.242·17-s − 1.37·19-s − 1.25·23-s + 1.73·27-s − 1.67·29-s + 0.718·31-s − 2.61·33-s − 0.328·37-s − 1.44·39-s + 0.624·41-s − 1.52·43-s − 0.145·47-s − 0.420·51-s − 0.549·53-s − 2.38·57-s + 1.04·59-s + 1.02·61-s − 1.46·67-s − 2.16·69-s + 0.949·71-s + 0.234·73-s + 1.46·79-s + 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 + T + p T^{2} \) | 1.17.b |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 6 T + p T^{2} \) | 1.23.g |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 8 T + p T^{2} \) | 1.61.ai |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−8.139418361854724874623097588421, −7.44099969728554811267307044223, −6.73090257323593090666183314764, −5.65281322329375920967173457432, −4.73855020505789086897581548645, −4.01006359968701985097827443222, −3.18951693575181046626973642778, −2.29908472484109817747562649632, −1.98802964785301257384067880121, 0,
1.98802964785301257384067880121, 2.29908472484109817747562649632, 3.18951693575181046626973642778, 4.01006359968701985097827443222, 4.73855020505789086897581548645, 5.65281322329375920967173457432, 6.73090257323593090666183314764, 7.44099969728554811267307044223, 8.139418361854724874623097588421
