| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s − 8-s + 9-s + 10-s − 2·11-s + 2·12-s + 6·13-s − 2·15-s + 16-s − 17-s − 18-s + 8·19-s − 20-s + 2·22-s − 2·23-s − 2·24-s + 25-s − 6·26-s − 4·27-s + 6·29-s + 2·30-s + 2·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.577·12-s + 1.66·13-s − 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 1.83·19-s − 0.223·20-s + 0.426·22-s − 0.417·23-s − 0.408·24-s + 1/5·25-s − 1.17·26-s − 0.769·27-s + 1.11·29-s + 0.365·30-s + 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.200299754\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.200299754\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 - 2 T + p T^{2} \) | 1.3.ac |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−8.092484353502950576683404302437, −7.42717455099730047106034522764, −6.61079142784416346954959499226, −5.88360940200955664519328599726, −5.00461346323319973754488181716, −3.95114382547935376964949865766, −3.25494214016387779422541683820, −2.78847960718251599716771509585, −1.71450635227290170426580654073, −0.78804658243269606528154158871,
0.78804658243269606528154158871, 1.71450635227290170426580654073, 2.78847960718251599716771509585, 3.25494214016387779422541683820, 3.95114382547935376964949865766, 5.00461346323319973754488181716, 5.88360940200955664519328599726, 6.61079142784416346954959499226, 7.42717455099730047106034522764, 8.092484353502950576683404302437
