| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 2·7-s + 8-s + 9-s − 12-s + 2·14-s + 16-s + 2·17-s + 18-s − 2·19-s − 2·21-s − 24-s − 27-s + 2·28-s − 6·29-s + 32-s + 2·34-s + 36-s + 6·37-s − 2·38-s + 6·41-s − 2·42-s + 2·43-s + 4·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.534·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.458·19-s − 0.436·21-s − 0.204·24-s − 0.192·27-s + 0.377·28-s − 1.11·29-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.986·37-s − 0.324·38-s + 0.937·41-s − 0.308·42-s + 0.304·43-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.261781120\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.261781120\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 4 T + p T^{2} \) | 1.47.ae |
| 53 | \( 1 + 2 T + p T^{2} \) | 1.53.c |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 - 2 T + p T^{2} \) | 1.79.ac |
| 83 | \( 1 - 8 T + p T^{2} \) | 1.83.ai |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 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
−15.71257071743029, −15.11587361363290, −14.77054462237178, −14.09759530455769, −13.69616386524875, −12.86087139913826, −12.57006464745956, −11.97345248358275, −11.27073891628065, −11.01549370540200, −10.47084951117836, −9.605608313703346, −9.179681903503014, −8.092945545759616, −7.835025233580815, −7.086500839646782, −6.412193586829063, −5.789863771845004, −5.301220350233838, −4.596224832082964, −4.092448527108023, −3.312916824380598, −2.368748504665450, −1.665841687231654, −0.7141236717834001,
0.7141236717834001, 1.665841687231654, 2.368748504665450, 3.312916824380598, 4.092448527108023, 4.596224832082964, 5.301220350233838, 5.789863771845004, 6.412193586829063, 7.086500839646782, 7.835025233580815, 8.092945545759616, 9.179681903503014, 9.605608313703346, 10.47084951117836, 11.01549370540200, 11.27073891628065, 11.97345248358275, 12.57006464745956, 12.86087139913826, 13.69616386524875, 14.09759530455769, 14.77054462237178, 15.11587361363290, 15.71257071743029
