| L(s) = 1 | + 3-s − 5·7-s + 9-s + 2·11-s − 2·13-s + 4·17-s − 19-s − 5·21-s − 2·23-s + 27-s + 9·29-s − 8·31-s + 2·33-s + 2·37-s − 2·39-s + 7·41-s − 12·43-s − 12·47-s + 18·49-s + 4·51-s + 9·53-s − 57-s + 15·59-s − 13·61-s − 5·63-s + 8·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s + 0.603·11-s − 0.554·13-s + 0.970·17-s − 0.229·19-s − 1.09·21-s − 0.417·23-s + 0.192·27-s + 1.67·29-s − 1.43·31-s + 0.348·33-s + 0.328·37-s − 0.320·39-s + 1.09·41-s − 1.82·43-s − 1.75·47-s + 18/7·49-s + 0.560·51-s + 1.23·53-s − 0.132·57-s + 1.95·59-s − 1.66·61-s − 0.629·63-s + 0.977·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.730248043\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.730248043\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 + 5 T + p T^{2} \) | 1.7.f |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 - 15 T + p T^{2} \) | 1.59.ap |
| 61 | \( 1 + 13 T + p T^{2} \) | 1.61.n |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - T + p T^{2} \) | 1.71.ab |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−8.179054080484572184115837116638, −7.31791495484327663216122035836, −6.71102180781504964835397559992, −6.18069926162879974807742981379, −5.30536925208551409310299219643, −4.25718735462672512799554873907, −3.45684034972003434249021202865, −3.02646480247497283660231212183, −2.01776924708312378077896558175, −0.66008638117534687638613914589,
0.66008638117534687638613914589, 2.01776924708312378077896558175, 3.02646480247497283660231212183, 3.45684034972003434249021202865, 4.25718735462672512799554873907, 5.30536925208551409310299219643, 6.18069926162879974807742981379, 6.71102180781504964835397559992, 7.31791495484327663216122035836, 8.179054080484572184115837116638
