| L(s) = 1 | + 2·2-s + 2·4-s + 3·7-s − 3·9-s + 4·11-s + 2·13-s + 6·14-s − 4·16-s + 8·17-s − 6·18-s − 3·19-s + 8·22-s + 3·23-s + 4·26-s + 6·28-s + 7·29-s − 10·31-s − 8·32-s + 16·34-s − 6·36-s − 7·37-s − 6·38-s + 9·41-s − 43-s + 8·44-s + 6·46-s + 11·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.13·7-s − 9-s + 1.20·11-s + 0.554·13-s + 1.60·14-s − 16-s + 1.94·17-s − 1.41·18-s − 0.688·19-s + 1.70·22-s + 0.625·23-s + 0.784·26-s + 1.13·28-s + 1.29·29-s − 1.79·31-s − 1.41·32-s + 2.74·34-s − 36-s − 1.15·37-s − 0.973·38-s + 1.40·41-s − 0.152·43-s + 1.20·44-s + 0.884·46-s + 1.60·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.071009002\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.071009002\) |
| \(L(\frac{3}{2})\) |
|
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 | 5 | \( 1 \) | |
| 197 | \( 1 - T \) | |
| good | 2 | \( 1 - p T + p T^{2} \) | 1.2.ac |
| 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 + 3 T + p T^{2} \) | 1.19.d |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 7 T + p T^{2} \) | 1.29.ah |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 7 T + p T^{2} \) | 1.37.h |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 11 T + p T^{2} \) | 1.47.al |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 7 T + p T^{2} \) | 1.83.ah |
| 89 | \( 1 + 8 T + p T^{2} \) | 1.89.i |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−8.274186634151122597643469974243, −7.38321960705366179874531542523, −6.54013718671721998420560379289, −5.78671884243827565264966890450, −5.35946416359992619882619443150, −4.58640504680846064928838550970, −3.74621326726348715367743360170, −3.23747729221580895986788159609, −2.14457455330861137661769914181, −1.06964920838376654214460855629,
1.06964920838376654214460855629, 2.14457455330861137661769914181, 3.23747729221580895986788159609, 3.74621326726348715367743360170, 4.58640504680846064928838550970, 5.35946416359992619882619443150, 5.78671884243827565264966890450, 6.54013718671721998420560379289, 7.38321960705366179874531542523, 8.274186634151122597643469974243
