| L(s) = 1 | − 2·7-s − 3·9-s + 11-s − 6·13-s − 2·17-s + 4·19-s − 4·23-s − 6·29-s + 8·31-s + 8·37-s − 2·41-s − 10·43-s + 12·47-s − 3·49-s + 8·53-s + 2·61-s + 6·63-s + 4·67-s + 4·71-s − 10·73-s − 2·77-s + 16·79-s + 9·81-s + 6·83-s + 2·89-s + 12·91-s − 4·97-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 9-s + 0.301·11-s − 1.66·13-s − 0.485·17-s + 0.917·19-s − 0.834·23-s − 1.11·29-s + 1.43·31-s + 1.31·37-s − 0.312·41-s − 1.52·43-s + 1.75·47-s − 3/7·49-s + 1.09·53-s + 0.256·61-s + 0.755·63-s + 0.488·67-s + 0.474·71-s − 1.17·73-s − 0.227·77-s + 1.80·79-s + 81-s + 0.658·83-s + 0.211·89-s + 1.25·91-s − 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.030367792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030367792\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 10 T + p T^{2} \) | 1.43.k |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 8 T + p T^{2} \) | 1.53.ai |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 4 T + p T^{2} \) | 1.97.e |
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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
−8.316749043615038782113762357571, −7.63400276553910317378609970000, −6.89820462617537603697690597811, −6.17099498878514395085869399352, −5.45148617934067740987373851308, −4.68613138727415837961830150039, −3.71248749438835835772448329778, −2.83959592654667806458489767088, −2.18146285515880200411456980651, −0.53684463213556938705685656778,
0.53684463213556938705685656778, 2.18146285515880200411456980651, 2.83959592654667806458489767088, 3.71248749438835835772448329778, 4.68613138727415837961830150039, 5.45148617934067740987373851308, 6.17099498878514395085869399352, 6.89820462617537603697690597811, 7.63400276553910317378609970000, 8.316749043615038782113762357571
