| L(s) = 1 | − 3-s − 2·5-s − 4·7-s + 9-s + 13-s + 2·15-s − 4·17-s + 6·19-s + 4·21-s + 6·23-s − 25-s − 27-s − 6·29-s + 6·31-s + 8·35-s − 4·37-s − 39-s + 10·41-s + 8·43-s − 2·45-s − 12·47-s + 9·49-s + 4·51-s − 12·53-s − 6·57-s + 12·59-s − 14·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s + 0.277·13-s + 0.516·15-s − 0.970·17-s + 1.37·19-s + 0.872·21-s + 1.25·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.07·31-s + 1.35·35-s − 0.657·37-s − 0.160·39-s + 1.56·41-s + 1.21·43-s − 0.298·45-s − 1.75·47-s + 9/7·49-s + 0.560·51-s − 1.64·53-s − 0.794·57-s + 1.56·59-s − 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 151008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.055215234\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.055215234\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 12 T + p T^{2} \) | 1.53.m |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.12413687896350, −12.82668039317215, −12.47316893013631, −11.86740208444848, −11.38078441061069, −11.04618916107998, −10.66087981310296, −9.758588392625808, −9.599960055024864, −9.166883414601023, −8.513992199151616, −7.856186615620831, −7.412558053507887, −6.913738347931455, −6.510517130734491, −5.965043789634480, −5.483955432829047, −4.740045283671179, −4.349167078828301, −3.555918746139317, −3.316036718437376, −2.709053545237679, −1.840569300990638, −0.8696617086955673, −0.4140621387389451,
0.4140621387389451, 0.8696617086955673, 1.840569300990638, 2.709053545237679, 3.316036718437376, 3.555918746139317, 4.349167078828301, 4.740045283671179, 5.483955432829047, 5.965043789634480, 6.510517130734491, 6.913738347931455, 7.412558053507887, 7.856186615620831, 8.513992199151616, 9.166883414601023, 9.599960055024864, 9.758588392625808, 10.66087981310296, 11.04618916107998, 11.38078441061069, 11.86740208444848, 12.47316893013631, 12.82668039317215, 13.12413687896350
