| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s + 6·13-s − 15-s + 2·17-s − 19-s + 4·21-s + 8·23-s + 25-s + 27-s − 2·29-s − 8·31-s − 4·35-s − 10·37-s + 6·39-s + 6·41-s + 8·43-s − 45-s + 9·49-s + 2·51-s + 2·53-s − 57-s + 12·59-s + 2·61-s + 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s + 1.66·13-s − 0.258·15-s + 0.485·17-s − 0.229·19-s + 0.872·21-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.676·35-s − 1.64·37-s + 0.960·39-s + 0.937·41-s + 1.21·43-s − 0.149·45-s + 9/7·49-s + 0.280·51-s + 0.274·53-s − 0.132·57-s + 1.56·59-s + 0.256·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.005867679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.005867679\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 + T \) | |
| 19 | \( 1 + T \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−15.80124086816435, −15.11173138134170, −14.65761441680613, −14.26464553694087, −13.72641906045483, −12.91680479992787, −12.73540470894392, −11.68651428780026, −11.35397124708215, −10.77810647323104, −10.48024300793645, −9.350766028109867, −8.810060865388057, −8.516608581292268, −7.870615390547837, −7.302598553864416, −6.789826743804290, −5.693064344627712, −5.314263358523021, −4.479911367118179, −3.830438625396385, −3.341662631157321, −2.322497773234736, −1.516768647459630, −0.9072503016427667,
0.9072503016427667, 1.516768647459630, 2.322497773234736, 3.341662631157321, 3.830438625396385, 4.479911367118179, 5.314263358523021, 5.693064344627712, 6.789826743804290, 7.302598553864416, 7.870615390547837, 8.516608581292268, 8.810060865388057, 9.350766028109867, 10.48024300793645, 10.77810647323104, 11.35397124708215, 11.68651428780026, 12.73540470894392, 12.91680479992787, 13.72641906045483, 14.26464553694087, 14.65761441680613, 15.11173138134170, 15.80124086816435
