| L(s) = 1 | + 7-s − 2·11-s − 4·13-s + 6·17-s + 6·19-s − 8·23-s + 2·29-s + 10·31-s − 2·37-s − 10·41-s + 4·43-s − 8·47-s + 49-s + 4·53-s + 8·59-s + 6·61-s − 12·67-s + 6·71-s + 12·73-s − 2·77-s − 8·79-s − 4·83-s + 10·89-s − 4·91-s − 8·97-s − 6·101-s + 16·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 0.603·11-s − 1.10·13-s + 1.45·17-s + 1.37·19-s − 1.66·23-s + 0.371·29-s + 1.79·31-s − 0.328·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.549·53-s + 1.04·59-s + 0.768·61-s − 1.46·67-s + 0.712·71-s + 1.40·73-s − 0.227·77-s − 0.900·79-s − 0.439·83-s + 1.05·89-s − 0.419·91-s − 0.812·97-s − 0.597·101-s + 1.57·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.900460998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.900460998\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 10 T + p T^{2} \) | 1.31.ak |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 - 12 T + p T^{2} \) | 1.73.am |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−7.971034203992665878044705193594, −7.50133793506589037574542692451, −6.70426937681704147074954471218, −5.74977630435349863381252332604, −5.20373160853444557562739913721, −4.56710095336295679219255023234, −3.51538871880217009379501983822, −2.79976216618028503900862442016, −1.86141066312668178729605899630, −0.71904722085841542845310369754,
0.71904722085841542845310369754, 1.86141066312668178729605899630, 2.79976216618028503900862442016, 3.51538871880217009379501983822, 4.56710095336295679219255023234, 5.20373160853444557562739913721, 5.74977630435349863381252332604, 6.70426937681704147074954471218, 7.50133793506589037574542692451, 7.971034203992665878044705193594
