| L(s) = 1 | + 2·5-s + 3·7-s + 2·11-s + 13-s + 4·17-s − 19-s + 8·23-s − 25-s − 6·29-s − 7·31-s + 6·35-s + 3·37-s − 6·41-s + 7·43-s − 6·47-s + 2·49-s + 14·53-s + 4·55-s + 2·59-s − 14·61-s + 2·65-s + 8·67-s + 10·71-s − 14·73-s + 6·77-s − 5·79-s − 8·83-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.13·7-s + 0.603·11-s + 0.277·13-s + 0.970·17-s − 0.229·19-s + 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.25·31-s + 1.01·35-s + 0.493·37-s − 0.937·41-s + 1.06·43-s − 0.875·47-s + 2/7·49-s + 1.92·53-s + 0.539·55-s + 0.260·59-s − 1.79·61-s + 0.248·65-s + 0.977·67-s + 1.18·71-s − 1.63·73-s + 0.683·77-s − 0.562·79-s − 0.878·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.592445600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.592445600\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 37 | \( 1 - 3 T + p T^{2} \) | 1.37.ad |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 - 2 T + p T^{2} \) | 1.59.ac |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−9.194626474433933678994403889035, −8.512515083817511118843218762747, −7.59207667107829549324270725650, −6.89711166759413012889381362916, −5.77600466769903506016105639686, −5.34751294951741050403865556943, −4.33177872683012152789172947335, −3.29112668439042487815688119078, −2.00958031561816070563221601703, −1.23476617459819810203195540994,
1.23476617459819810203195540994, 2.00958031561816070563221601703, 3.29112668439042487815688119078, 4.33177872683012152789172947335, 5.34751294951741050403865556943, 5.77600466769903506016105639686, 6.89711166759413012889381362916, 7.59207667107829549324270725650, 8.512515083817511118843218762747, 9.194626474433933678994403889035
