| L(s) = 1 | + 3-s − 2·4-s − 4·7-s − 2·9-s − 6·11-s − 2·12-s + 13-s + 4·16-s + 6·17-s − 4·19-s − 4·21-s + 3·23-s − 5·27-s + 8·28-s − 3·29-s − 4·31-s − 6·33-s + 4·36-s + 2·37-s + 39-s + 6·41-s − 7·43-s + 12·44-s + 4·48-s + 9·49-s + 6·51-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s − 1.51·7-s − 2/3·9-s − 1.80·11-s − 0.577·12-s + 0.277·13-s + 16-s + 1.45·17-s − 0.917·19-s − 0.872·21-s + 0.625·23-s − 0.962·27-s + 1.51·28-s − 0.557·29-s − 0.718·31-s − 1.04·33-s + 2/3·36-s + 0.328·37-s + 0.160·39-s + 0.937·41-s − 1.06·43-s + 1.80·44-s + 0.577·48-s + 9/7·49-s + 0.840·51-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 2 | \( 1 + p T^{2} \) | 1.2.a |
| 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 11 | \( 1 + 6 T + p T^{2} \) | 1.11.g |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 9 T + p T^{2} \) | 1.53.j |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 + 6 T + p T^{2} \) | 1.71.g |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 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
−10.90393326812070592888171831546, −9.984664310915636855598086053167, −9.326012139266372897443285677601, −8.361693078894093264166227718292, −7.60238633119367620508279222313, −6.03400631967558484660978662749, −5.20181778268865928931694844005, −3.60747185046759420693176083431, −2.83968371298038180105486860299, 0,
2.83968371298038180105486860299, 3.60747185046759420693176083431, 5.20181778268865928931694844005, 6.03400631967558484660978662749, 7.60238633119367620508279222313, 8.361693078894093264166227718292, 9.326012139266372897443285677601, 9.984664310915636855598086053167, 10.90393326812070592888171831546
