| L(s) = 1 | − 3-s + 5-s + 9-s − 2·13-s − 15-s + 6·17-s − 4·23-s + 25-s − 27-s + 2·29-s + 8·31-s − 6·37-s + 2·39-s + 6·41-s − 12·43-s + 45-s + 12·47-s − 6·51-s + 10·53-s + 8·59-s − 10·61-s − 2·65-s + 12·67-s + 4·69-s + 8·71-s − 10·73-s − 75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1/3·9-s − 0.554·13-s − 0.258·15-s + 1.45·17-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 1.43·31-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 1.82·43-s + 0.149·45-s + 1.75·47-s − 0.840·51-s + 1.37·53-s + 1.04·59-s − 1.28·61-s − 0.248·65-s + 1.46·67-s + 0.481·69-s + 0.949·71-s − 1.17·73-s − 0.115·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.180418666\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.180418666\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 6 T + p T^{2} \) | 1.37.g |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 12 T + p T^{2} \) | 1.43.m |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 10 T + p T^{2} \) | 1.61.k |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + 18 T + p T^{2} \) | 1.97.s |
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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
−14.58164674187140, −13.90844878156037, −13.74914570234523, −13.01879973609270, −12.34851481276749, −11.95628078870494, −11.81357778608161, −10.80409860561501, −10.40078159850891, −9.989488892764292, −9.543582330636655, −8.871477307457646, −8.107967476419953, −7.796427811980465, −6.997268873652654, −6.560379872642531, −5.931149260594388, −5.336356273623024, −5.021640021086029, −4.158836116291951, −3.603837962831549, −2.753756587965525, −2.149368125857590, −1.257892369298558, −0.5911141244024554,
0.5911141244024554, 1.257892369298558, 2.149368125857590, 2.753756587965525, 3.603837962831549, 4.158836116291951, 5.021640021086029, 5.336356273623024, 5.931149260594388, 6.560379872642531, 6.997268873652654, 7.796427811980465, 8.107967476419953, 8.871477307457646, 9.543582330636655, 9.989488892764292, 10.40078159850891, 10.80409860561501, 11.81357778608161, 11.95628078870494, 12.34851481276749, 13.01879973609270, 13.74914570234523, 13.90844878156037, 14.58164674187140
