| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 4·13-s − 14-s + 16-s − 2·17-s − 6·19-s − 20-s + 4·23-s − 4·25-s − 4·26-s + 28-s + 8·29-s − 2·31-s − 32-s + 2·34-s − 35-s + 37-s + 6·38-s + 40-s − 41-s − 4·43-s − 4·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s − 1.37·19-s − 0.223·20-s + 0.834·23-s − 4/5·25-s − 0.784·26-s + 0.188·28-s + 1.48·29-s − 0.359·31-s − 0.176·32-s + 0.342·34-s − 0.169·35-s + 0.164·37-s + 0.973·38-s + 0.158·40-s − 0.156·41-s − 0.609·43-s − 0.589·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45738 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| good | 5 | \( 1 + T + p T^{2} \) | 1.5.b |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + T + p T^{2} \) | 1.41.b |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 9 T + p T^{2} \) | 1.59.j |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 - 14 T + p T^{2} \) | 1.73.ao |
| 79 | \( 1 + 3 T + p T^{2} \) | 1.79.d |
| 83 | \( 1 + 9 T + p T^{2} \) | 1.83.j |
| 89 | \( 1 + 15 T + p T^{2} \) | 1.89.p |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−15.19368954321157, −14.38970379346031, −13.89504384390755, −13.31082261934393, −12.77027878441841, −12.24783471450344, −11.54844617154450, −11.27119990396553, −10.66133305161745, −10.35728209507741, −9.613113115695960, −8.934951402761016, −8.468506572519074, −8.287535640588366, −7.467315861127866, −7.005384079818644, −6.205290608834863, −6.072408070575985, −4.992586754791590, −4.485475494879266, −3.808578649926965, −3.166662220174916, −2.354724475039825, −1.679314896902835, −0.9177761551729414, 0,
0.9177761551729414, 1.679314896902835, 2.354724475039825, 3.166662220174916, 3.808578649926965, 4.485475494879266, 4.992586754791590, 6.072408070575985, 6.205290608834863, 7.005384079818644, 7.467315861127866, 8.287535640588366, 8.468506572519074, 8.934951402761016, 9.613113115695960, 10.35728209507741, 10.66133305161745, 11.27119990396553, 11.54844617154450, 12.24783471450344, 12.77027878441841, 13.31082261934393, 13.89504384390755, 14.38970379346031, 15.19368954321157
