| L(s) = 1 | − 2-s − 4-s + 4·7-s + 3·8-s + 4·11-s + 2·13-s − 4·14-s − 16-s + 6·17-s − 6·19-s − 4·22-s − 8·23-s − 2·26-s − 4·28-s + 6·29-s + 2·31-s − 5·32-s − 6·34-s + 37-s + 6·38-s + 10·43-s − 4·44-s + 8·46-s + 12·47-s + 9·49-s − 2·52-s − 4·53-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.51·7-s + 1.06·8-s + 1.20·11-s + 0.554·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s − 1.37·19-s − 0.852·22-s − 1.66·23-s − 0.392·26-s − 0.755·28-s + 1.11·29-s + 0.359·31-s − 0.883·32-s − 1.02·34-s + 0.164·37-s + 0.973·38-s + 1.52·43-s − 0.603·44-s + 1.17·46-s + 1.75·47-s + 9/7·49-s − 0.277·52-s − 0.549·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.818664648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.818664648\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + 8 T + p T^{2} \) | 1.23.i |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 + 4 T + p T^{2} \) | 1.53.e |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.989134293206019422099843447701, −7.45931834754504439118953033090, −6.41407838035907785899025449597, −5.77649029118277826087547800935, −4.91776466362877789663015745428, −4.14037475233003146585035603427, −3.87384064478032847814975130319, −2.32753203651138879809455487505, −1.46292789360955583924902668922, −0.850977005362306578443565050761,
0.850977005362306578443565050761, 1.46292789360955583924902668922, 2.32753203651138879809455487505, 3.87384064478032847814975130319, 4.14037475233003146585035603427, 4.91776466362877789663015745428, 5.77649029118277826087547800935, 6.41407838035907785899025449597, 7.45931834754504439118953033090, 7.989134293206019422099843447701
