| L(s) = 1 | + 2·7-s − 6·11-s − 13-s + 19-s + 6·23-s + 3·29-s + 8·31-s + 37-s − 9·41-s − 8·43-s − 3·47-s − 3·49-s − 3·53-s + 6·59-s + 10·61-s + 13·67-s + 9·71-s − 4·73-s − 12·77-s + 11·79-s − 12·83-s − 6·89-s − 2·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.80·11-s − 0.277·13-s + 0.229·19-s + 1.25·23-s + 0.557·29-s + 1.43·31-s + 0.164·37-s − 1.40·41-s − 1.21·43-s − 0.437·47-s − 3/7·49-s − 0.412·53-s + 0.781·59-s + 1.28·61-s + 1.58·67-s + 1.06·71-s − 0.468·73-s − 1.36·77-s + 1.23·79-s − 1.31·83-s − 0.635·89-s − 0.209·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.034233490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.034233490\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−13.08397088072420, −12.80171442132459, −12.15382751716536, −11.67010452173621, −11.20866412646871, −10.84721673949750, −10.22368206089553, −9.927425436160987, −9.492103994670448, −8.527495907421245, −8.347756239058381, −8.008490268317246, −7.431580814640011, −6.700877908173978, −6.618887791239840, −5.536999743591046, −5.177322905442943, −4.956357405387216, −4.376972917761239, −3.548090354134731, −2.935352009793031, −2.560903737667742, −1.890313732759803, −1.152899667876911, −0.4278523290622178,
0.4278523290622178, 1.152899667876911, 1.890313732759803, 2.560903737667742, 2.935352009793031, 3.548090354134731, 4.376972917761239, 4.956357405387216, 5.177322905442943, 5.536999743591046, 6.618887791239840, 6.700877908173978, 7.431580814640011, 8.008490268317246, 8.347756239058381, 8.527495907421245, 9.492103994670448, 9.927425436160987, 10.22368206089553, 10.84721673949750, 11.20866412646871, 11.67010452173621, 12.15382751716536, 12.80171442132459, 13.08397088072420
