| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s + 4·11-s + 13-s + 16-s + 6·17-s + 20-s − 4·22-s + 2·23-s + 25-s − 26-s + 6·29-s + 2·31-s − 32-s − 6·34-s − 6·37-s − 40-s − 2·41-s − 8·43-s + 4·44-s − 2·46-s − 8·47-s − 50-s + 52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s + 1.20·11-s + 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.223·20-s − 0.852·22-s + 0.417·23-s + 1/5·25-s − 0.196·26-s + 1.11·29-s + 0.359·31-s − 0.176·32-s − 1.02·34-s − 0.986·37-s − 0.158·40-s − 0.312·41-s − 1.21·43-s + 0.603·44-s − 0.294·46-s − 1.16·47-s − 0.141·50-s + 0.138·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.359324304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.359324304\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 2 T + p T^{2} \) | 1.89.ac |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−14.47252928115987, −13.81854215876865, −13.54701278067312, −12.68164597685299, −12.15586090012945, −11.90363981058521, −11.26826706744091, −10.67749173701915, −10.14428117349192, −9.736661375641352, −9.279389637456876, −8.624070300349291, −8.294253630083041, −7.618983847471247, −7.002483440229518, −6.407903619111915, −6.174550680350494, −5.269271019721089, −4.860710929001752, −3.923527049966874, −3.301277376834696, −2.843364354592383, −1.734191101035776, −1.408858736237940, −0.6278675013501225,
0.6278675013501225, 1.408858736237940, 1.734191101035776, 2.843364354592383, 3.301277376834696, 3.923527049966874, 4.860710929001752, 5.269271019721089, 6.174550680350494, 6.407903619111915, 7.002483440229518, 7.618983847471247, 8.294253630083041, 8.624070300349291, 9.279389637456876, 9.736661375641352, 10.14428117349192, 10.67749173701915, 11.26826706744091, 11.90363981058521, 12.15586090012945, 12.68164597685299, 13.54701278067312, 13.81854215876865, 14.47252928115987
