| L(s) = 1 | + 2·7-s − 5·11-s − 13-s + 2·17-s + 8·19-s − 7·23-s − 4·29-s + 2·31-s + 9·37-s − 6·41-s − 8·43-s − 47-s − 3·49-s − 59-s + 7·61-s − 10·67-s + 3·71-s + 6·73-s − 10·77-s − 6·79-s + 16·83-s + 6·89-s − 2·91-s − 97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 1.50·11-s − 0.277·13-s + 0.485·17-s + 1.83·19-s − 1.45·23-s − 0.742·29-s + 0.359·31-s + 1.47·37-s − 0.937·41-s − 1.21·43-s − 0.145·47-s − 3/7·49-s − 0.130·59-s + 0.896·61-s − 1.22·67-s + 0.356·71-s + 0.702·73-s − 1.13·77-s − 0.675·79-s + 1.75·83-s + 0.635·89-s − 0.209·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + T + p T^{2} \) | 1.13.b |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 7 T + p T^{2} \) | 1.23.h |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + T + p T^{2} \) | 1.59.b |
| 61 | \( 1 - 7 T + p T^{2} \) | 1.61.ah |
| 67 | \( 1 + 10 T + p T^{2} \) | 1.67.k |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 - 6 T + p T^{2} \) | 1.73.ag |
| 79 | \( 1 + 6 T + p T^{2} \) | 1.79.g |
| 83 | \( 1 - 16 T + p T^{2} \) | 1.83.aq |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 + T + p T^{2} \) | 1.97.b |
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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
−16.66687400191135, −16.18174928134170, −15.80969531001992, −14.95531756824014, −14.70155347654525, −13.77896706494968, −13.55240333895909, −12.85774954844322, −12.02472642886396, −11.72020173925840, −11.05198982569210, −10.32793307181727, −9.850606850354555, −9.335604678662710, −8.205593898554778, −7.939546074427177, −7.520528262065075, −6.616982051169461, −5.671396807729341, −5.264410025204722, −4.687417204878423, −3.707503224542207, −2.943543826259050, −2.164056402735020, −1.243188414356886, 0,
1.243188414356886, 2.164056402735020, 2.943543826259050, 3.707503224542207, 4.687417204878423, 5.264410025204722, 5.671396807729341, 6.616982051169461, 7.520528262065075, 7.939546074427177, 8.205593898554778, 9.335604678662710, 9.850606850354555, 10.32793307181727, 11.05198982569210, 11.72020173925840, 12.02472642886396, 12.85774954844322, 13.55240333895909, 13.77896706494968, 14.70155347654525, 14.95531756824014, 15.80969531001992, 16.18174928134170, 16.66687400191135
