| L(s) = 1 | − 2-s + 4-s + 7-s − 8-s − 13-s − 14-s + 16-s − 2·19-s − 5·25-s + 26-s + 28-s − 2·29-s + 2·31-s − 32-s − 10·37-s + 2·38-s + 4·41-s + 8·43-s − 10·47-s + 49-s + 5·50-s − 52-s + 6·53-s − 56-s + 2·58-s + 2·61-s − 2·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.458·19-s − 25-s + 0.196·26-s + 0.188·28-s − 0.371·29-s + 0.359·31-s − 0.176·32-s − 1.64·37-s + 0.324·38-s + 0.624·41-s + 1.21·43-s − 1.45·47-s + 1/7·49-s + 0.707·50-s − 0.138·52-s + 0.824·53-s − 0.133·56-s + 0.262·58-s + 0.256·61-s − 0.254·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198198 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.047572185\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.047572185\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 10 T + p T^{2} \) | 1.47.k |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 12 T + p T^{2} \) | 1.67.m |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 + 8 T + p T^{2} \) | 1.73.i |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−13.05433880873872, −12.42799045228802, −12.13897186108451, −11.51568819207162, −11.24073874445711, −10.61991574958666, −10.25817335833335, −9.791527936430220, −9.243142607157601, −8.826874707810521, −8.348937665451812, −7.815232770847047, −7.457470362055028, −6.932051196250995, −6.339996201410600, −5.907363150041262, −5.280331195379391, −4.813276503637176, −4.093586316239841, −3.639464067031942, −2.936965843961931, −2.243383130154787, −1.857805692262920, −1.140250204893639, −0.3396417157734687,
0.3396417157734687, 1.140250204893639, 1.857805692262920, 2.243383130154787, 2.936965843961931, 3.639464067031942, 4.093586316239841, 4.813276503637176, 5.280331195379391, 5.907363150041262, 6.339996201410600, 6.932051196250995, 7.457470362055028, 7.815232770847047, 8.348937665451812, 8.826874707810521, 9.243142607157601, 9.791527936430220, 10.25817335833335, 10.61991574958666, 11.24073874445711, 11.51568819207162, 12.13897186108451, 12.42799045228802, 13.05433880873872
