| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 3·7-s − 8-s + 9-s + 3·11-s + 12-s + 3·14-s + 16-s + 3·17-s − 18-s − 3·21-s − 3·22-s + 4·23-s − 24-s + 27-s − 3·28-s + 5·29-s + 3·31-s − 32-s + 3·33-s − 3·34-s + 36-s + 12·37-s − 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.904·11-s + 0.288·12-s + 0.801·14-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.654·21-s − 0.639·22-s + 0.834·23-s − 0.204·24-s + 0.192·27-s − 0.566·28-s + 0.928·29-s + 0.538·31-s − 0.176·32-s + 0.522·33-s − 0.514·34-s + 1/6·36-s + 1.97·37-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.155354989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155354989\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 5 | \( 1 \) | |
| 13 | \( 1 \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 5 T + p T^{2} \) | 1.29.af |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 + 3 T + p T^{2} \) | 1.47.d |
| 53 | \( 1 - 9 T + p T^{2} \) | 1.53.aj |
| 59 | \( 1 + 15 T + p T^{2} \) | 1.59.p |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 16 T + p T^{2} \) | 1.73.aq |
| 79 | \( 1 + p T^{2} \) | 1.79.a |
| 83 | \( 1 - T + p T^{2} \) | 1.83.ab |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.32654457217338, −14.97689472519538, −14.28375303280795, −13.81476933504325, −13.20766082705588, −12.52306229267742, −12.26675642127223, −11.49174376201822, −10.94885952016994, −10.21007000104575, −9.773491866289871, −9.288512891615707, −8.954322977531755, −8.098120318370264, −7.762375870768150, −6.901741047248674, −6.520369141466702, −6.028193068023840, −5.102779080036115, −4.250299641749215, −3.565798531406233, −2.959640454921020, −2.409052878958781, −1.298766109775751, −0.6971587212875826,
0.6971587212875826, 1.298766109775751, 2.409052878958781, 2.959640454921020, 3.565798531406233, 4.250299641749215, 5.102779080036115, 6.028193068023840, 6.520369141466702, 6.901741047248674, 7.762375870768150, 8.098120318370264, 8.954322977531755, 9.288512891615707, 9.773491866289871, 10.21007000104575, 10.94885952016994, 11.49174376201822, 12.26675642127223, 12.52306229267742, 13.20766082705588, 13.81476933504325, 14.28375303280795, 14.97689472519538, 15.32654457217338
