| L(s) = 1 | + 4·11-s − 2·13-s − 2·17-s − 8·19-s − 4·23-s + 6·29-s − 2·37-s + 6·41-s + 4·43-s + 12·47-s − 7·49-s − 6·53-s + 12·59-s + 14·61-s − 12·67-s − 2·73-s + 8·79-s + 4·83-s − 2·89-s + 14·97-s + 14·101-s + 8·103-s + 12·107-s + 14·109-s + 6·113-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 0.554·13-s − 0.485·17-s − 1.83·19-s − 0.834·23-s + 1.11·29-s − 0.328·37-s + 0.937·41-s + 0.609·43-s + 1.75·47-s − 49-s − 0.824·53-s + 1.56·59-s + 1.79·61-s − 1.46·67-s − 0.234·73-s + 0.900·79-s + 0.439·83-s − 0.211·89-s + 1.42·97-s + 1.39·101-s + 0.788·103-s + 1.16·107-s + 1.34·109-s + 0.564·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.800933544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.800933544\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| good | 7 | \( 1 + p T^{2} \) | 1.7.a |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 8 T + p T^{2} \) | 1.19.i |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−7.921476513679381029930942481143, −7.18147055562474599879969961526, −6.37965898072603990612496473676, −6.12426522257465007746358506675, −4.96471063433703571909091386089, −4.26589031535852789505659101099, −3.76818361104577650677990818112, −2.55076714240410676728377427283, −1.91531739888343518258290807796, −0.67022129765624857642580521738,
0.67022129765624857642580521738, 1.91531739888343518258290807796, 2.55076714240410676728377427283, 3.76818361104577650677990818112, 4.26589031535852789505659101099, 4.96471063433703571909091386089, 6.12426522257465007746358506675, 6.37965898072603990612496473676, 7.18147055562474599879969961526, 7.921476513679381029930942481143
