| L(s) = 1 | − 2-s + 4-s + 5-s − 7-s − 8-s − 10-s − 13-s + 14-s + 16-s − 4·19-s + 20-s + 4·23-s + 25-s + 26-s − 28-s − 2·29-s + 8·31-s − 32-s − 35-s + 2·37-s + 4·38-s − 40-s − 9·41-s + 7·43-s − 4·46-s + 47-s − 6·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s − 0.353·8-s − 0.316·10-s − 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.917·19-s + 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s − 0.371·29-s + 1.43·31-s − 0.176·32-s − 0.169·35-s + 0.328·37-s + 0.648·38-s − 0.158·40-s − 1.40·41-s + 1.06·43-s − 0.589·46-s + 0.145·47-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 141570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.324012520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.324012520\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 2 T + p T^{2} \) | 1.29.c |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 7 T + p T^{2} \) | 1.43.ah |
| 47 | \( 1 - T + p T^{2} \) | 1.47.ab |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 11 T + p T^{2} \) | 1.67.al |
| 71 | \( 1 + 10 T + p T^{2} \) | 1.71.k |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.11492972151418, −13.06455202590315, −12.47325202899694, −11.83377501809454, −11.49604673551776, −10.83315711409717, −10.48455571528628, −9.954356530088008, −9.591158090538225, −9.055966990495013, −8.618467690260066, −8.083412849129232, −7.638121874161112, −6.835755101191431, −6.663853623541260, −6.115494864392612, −5.490121882916357, −4.930055972868281, −4.334003313096491, −3.646256511249256, −2.925318293793161, −2.526260098441282, −1.832692519240303, −1.170726260840368, −0.4042657353875676,
0.4042657353875676, 1.170726260840368, 1.832692519240303, 2.526260098441282, 2.925318293793161, 3.646256511249256, 4.334003313096491, 4.930055972868281, 5.490121882916357, 6.115494864392612, 6.663853623541260, 6.835755101191431, 7.638121874161112, 8.083412849129232, 8.618467690260066, 9.055966990495013, 9.591158090538225, 9.954356530088008, 10.48455571528628, 10.83315711409717, 11.49604673551776, 11.83377501809454, 12.47325202899694, 13.06455202590315, 13.11492972151418
