| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s − 11-s + 2·14-s + 16-s − 6·17-s + 7·19-s − 20-s + 22-s + 3·23-s + 25-s − 2·28-s + 3·29-s − 8·31-s − 32-s + 6·34-s + 2·35-s − 2·37-s − 7·38-s + 40-s − 43-s − 44-s − 3·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 1.60·19-s − 0.223·20-s + 0.213·22-s + 0.625·23-s + 1/5·25-s − 0.377·28-s + 0.557·29-s − 1.43·31-s − 0.176·32-s + 1.02·34-s + 0.338·35-s − 0.328·37-s − 1.13·38-s + 0.158·40-s − 0.152·43-s − 0.150·44-s − 0.442·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 167310 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9894399490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9894399490\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 + T \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 17 | \( 1 + 6 T + p T^{2} \) | 1.17.g |
| 19 | \( 1 - 7 T + p T^{2} \) | 1.19.ah |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + T + p T^{2} \) | 1.43.b |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 6 T + p T^{2} \) | 1.59.ag |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 15 T + p T^{2} \) | 1.71.ap |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 9 T + p T^{2} \) | 1.83.aj |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 17 T + p T^{2} \) | 1.97.r |
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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
−13.24067750281716, −12.73599079832790, −12.20198677926595, −11.75275146197958, −11.27525013821426, −10.80974059952141, −10.45171059289495, −9.756808428667264, −9.397676950201454, −9.009126345649777, −8.455601495611362, −8.010126094277252, −7.307892148303366, −7.008946809815218, −6.664020138868197, −5.931578442648515, −5.340716581868148, −4.939850189232256, −4.082663349457730, −3.614377294391092, −3.023106459507613, −2.500383297486003, −1.843997906416556, −0.9738448103970154, −0.3827920051818086,
0.3827920051818086, 0.9738448103970154, 1.843997906416556, 2.500383297486003, 3.023106459507613, 3.614377294391092, 4.082663349457730, 4.939850189232256, 5.340716581868148, 5.931578442648515, 6.664020138868197, 7.008946809815218, 7.307892148303366, 8.010126094277252, 8.455601495611362, 9.009126345649777, 9.397676950201454, 9.756808428667264, 10.45171059289495, 10.80974059952141, 11.27525013821426, 11.75275146197958, 12.20198677926595, 12.73599079832790, 13.24067750281716
