| L(s) = 1 | − 3·9-s + 6·11-s + 4·13-s + 7·17-s − 4·19-s − 23-s + 6·29-s + 3·31-s + 4·37-s + 9·41-s − 8·43-s − 11·47-s + 4·53-s − 10·59-s + 2·61-s − 8·67-s − 3·71-s + 2·73-s − 17·79-s + 9·81-s + 2·83-s + 7·89-s − 97-s − 18·99-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 9-s + 1.80·11-s + 1.10·13-s + 1.69·17-s − 0.917·19-s − 0.208·23-s + 1.11·29-s + 0.538·31-s + 0.657·37-s + 1.40·41-s − 1.21·43-s − 1.60·47-s + 0.549·53-s − 1.30·59-s + 0.256·61-s − 0.977·67-s − 0.356·71-s + 0.234·73-s − 1.91·79-s + 81-s + 0.219·83-s + 0.741·89-s − 0.101·97-s − 1.80·99-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.175031109\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.175031109\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 7 T + p T^{2} \) | 1.17.ah |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 4 T + p T^{2} \) | 1.37.ae |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 11 T + p T^{2} \) | 1.47.l |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 + 10 T + p T^{2} \) | 1.59.k |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 + 3 T + p T^{2} \) | 1.71.d |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 17 T + p T^{2} \) | 1.79.r |
| 83 | \( 1 - 2 T + p T^{2} \) | 1.83.ac |
| 89 | \( 1 - 7 T + p T^{2} \) | 1.89.ah |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−13.93614960084548, −13.85926947868646, −12.88589474840124, −12.56406621570600, −11.86583230250079, −11.51180368829188, −11.30604209526617, −10.37132479221496, −10.12369006476040, −9.397560397002452, −8.871816409685668, −8.535554926173630, −8.001057935885731, −7.453480857955286, −6.509833373403232, −6.293440095899989, −5.931007925414550, −5.183676009097048, −4.412123192655633, −3.978497304662167, −3.222710517310009, −2.969457288775401, −1.856186357125840, −1.270889603558052, −0.6391274117039518,
0.6391274117039518, 1.270889603558052, 1.856186357125840, 2.969457288775401, 3.222710517310009, 3.978497304662167, 4.412123192655633, 5.183676009097048, 5.931007925414550, 6.293440095899989, 6.509833373403232, 7.453480857955286, 8.001057935885731, 8.535554926173630, 8.871816409685668, 9.397560397002452, 10.12369006476040, 10.37132479221496, 11.30604209526617, 11.51180368829188, 11.86583230250079, 12.56406621570600, 12.88589474840124, 13.85926947868646, 13.93614960084548
