| L(s) = 1 | − 2-s + 3·3-s + 4-s − 3·6-s − 8-s + 6·9-s − 3·11-s + 3·12-s − 4·13-s + 16-s + 3·17-s − 6·18-s + 5·19-s + 3·22-s + 4·23-s − 3·24-s + 4·26-s + 9·27-s − 8·31-s − 32-s − 9·33-s − 3·34-s + 6·36-s − 4·37-s − 5·38-s − 12·39-s + 11·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.353·8-s + 2·9-s − 0.904·11-s + 0.866·12-s − 1.10·13-s + 1/4·16-s + 0.727·17-s − 1.41·18-s + 1.14·19-s + 0.639·22-s + 0.834·23-s − 0.612·24-s + 0.784·26-s + 1.73·27-s − 1.43·31-s − 0.176·32-s − 1.56·33-s − 0.514·34-s + 36-s − 0.657·37-s − 0.811·38-s − 1.92·39-s + 1.71·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.766172318\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.766172318\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 79 | \( 1 - T \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 11 T + p T^{2} \) | 1.41.al |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 83 | \( 1 - 3 T + p T^{2} \) | 1.83.ad |
| 89 | \( 1 - 11 T + p T^{2} \) | 1.89.al |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−12.99204231376796, −12.83220320355571, −12.26294453979126, −11.65122872088888, −11.10645009073535, −10.55305660162405, −9.967440304932921, −9.671588033426837, −9.405907400732923, −8.792695400344188, −8.317825022965765, −7.886311705441099, −7.528159813833028, −7.075121596060296, −6.774568599483259, −5.675710861291354, −5.226478784423158, −4.831015083127810, −3.779620332964828, −3.551178342230027, −2.897624770093585, −2.418235078569965, −2.052325620137257, −1.243389264926036, −0.5579628015024135,
0.5579628015024135, 1.243389264926036, 2.052325620137257, 2.418235078569965, 2.897624770093585, 3.551178342230027, 3.779620332964828, 4.831015083127810, 5.226478784423158, 5.675710861291354, 6.774568599483259, 7.075121596060296, 7.528159813833028, 7.886311705441099, 8.317825022965765, 8.792695400344188, 9.405907400732923, 9.671588033426837, 9.967440304932921, 10.55305660162405, 11.10645009073535, 11.65122872088888, 12.26294453979126, 12.83220320355571, 12.99204231376796
