| L(s) = 1 | + 2-s + 3-s − 4-s + 5-s + 6-s − 4·7-s − 3·8-s + 9-s + 10-s − 12-s + 4·13-s − 4·14-s + 15-s − 16-s − 4·17-s + 18-s − 20-s − 4·21-s − 3·24-s + 25-s + 4·26-s + 27-s + 4·28-s + 8·29-s + 30-s − 8·31-s + 5·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 1.51·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s + 1.10·13-s − 1.06·14-s + 0.258·15-s − 1/4·16-s − 0.970·17-s + 0.235·18-s − 0.223·20-s − 0.872·21-s − 0.612·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.755·28-s + 1.48·29-s + 0.182·30-s − 1.43·31-s + 0.883·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 41 | \( 1 \) | |
| good | 2 | \( 1 - T + p T^{2} \) | 1.2.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 8 T + p T^{2} \) | 1.89.ai |
| 97 | \( 1 + 12 T + p T^{2} \) | 1.97.m |
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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
−15.46693861025939, −15.14630410577847, −14.34020658803396, −13.83551389611406, −13.54259255283765, −13.09535933308403, −12.57421959331773, −12.24359304598822, −11.35043289130466, −10.55571144335690, −10.21519766630436, −9.405550270047750, −8.978435450924956, −8.808030142866156, −7.938442994729359, −7.016598360021616, −6.528476065058694, −6.023127604678392, −5.478420612641659, −4.623934590751759, −3.965197188998412, −3.492030336035609, −2.894238418200664, −2.217938244522062, −1.044890511682053, 0,
1.044890511682053, 2.217938244522062, 2.894238418200664, 3.492030336035609, 3.965197188998412, 4.623934590751759, 5.478420612641659, 6.023127604678392, 6.528476065058694, 7.016598360021616, 7.938442994729359, 8.808030142866156, 8.978435450924956, 9.405550270047750, 10.21519766630436, 10.55571144335690, 11.35043289130466, 12.24359304598822, 12.57421959331773, 13.09535933308403, 13.54259255283765, 13.83551389611406, 14.34020658803396, 15.14630410577847, 15.46693861025939
