| L(s) = 1 | + 2·3-s + 3·5-s + 2·7-s + 9-s + 6·15-s + 3·17-s + 2·19-s + 4·21-s + 4·25-s − 4·27-s + 3·29-s − 8·31-s + 6·35-s + 11·37-s − 9·41-s − 4·43-s + 3·45-s + 6·47-s − 3·49-s + 6·51-s + 3·53-s + 4·57-s − 6·59-s + 7·61-s + 2·63-s − 14·67-s − 12·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.34·5-s + 0.755·7-s + 1/3·9-s + 1.54·15-s + 0.727·17-s + 0.458·19-s + 0.872·21-s + 4/5·25-s − 0.769·27-s + 0.557·29-s − 1.43·31-s + 1.01·35-s + 1.80·37-s − 1.40·41-s − 0.609·43-s + 0.447·45-s + 0.875·47-s − 3/7·49-s + 0.840·51-s + 0.412·53-s + 0.529·57-s − 0.781·59-s + 0.896·61-s + 0.251·63-s − 1.71·67-s − 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 327184 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 5 | \( 1 - 3 T + p T^{2} \) | 1.5.ad |
| 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 - 11 T + p T^{2} \) | 1.37.al |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 6 T + p T^{2} \) | 1.47.ag |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 - 7 T + p T^{2} \) | 1.73.ah |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 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
−13.04526160805695, −12.50884903490905, −11.86674099600288, −11.53306907186779, −10.95762742446780, −10.38274614865761, −10.02554077185978, −9.574728335764972, −9.155735749273571, −8.804601999981554, −8.261745549939626, −7.843187536956849, −7.413670220310034, −6.907047986169666, −6.167657450006980, −5.790079364913608, −5.368676152943877, −4.820896703266249, −4.271893467159255, −3.520655580188946, −3.173616313983413, −2.468924650351008, −2.196057049864122, −1.434642275288225, −1.234165872950038, 0,
1.234165872950038, 1.434642275288225, 2.196057049864122, 2.468924650351008, 3.173616313983413, 3.520655580188946, 4.271893467159255, 4.820896703266249, 5.368676152943877, 5.790079364913608, 6.167657450006980, 6.907047986169666, 7.413670220310034, 7.843187536956849, 8.261745549939626, 8.804601999981554, 9.155735749273571, 9.574728335764972, 10.02554077185978, 10.38274614865761, 10.95762742446780, 11.53306907186779, 11.86674099600288, 12.50884903490905, 13.04526160805695
