| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 12-s + 13-s + 14-s + 16-s + 4·17-s − 18-s + 21-s + 4·23-s + 24-s − 5·25-s − 26-s − 27-s − 28-s + 6·29-s + 2·31-s − 32-s − 4·34-s + 36-s + 8·37-s − 39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s − 0.235·18-s + 0.218·21-s + 0.834·23-s + 0.204·24-s − 25-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s − 0.685·34-s + 1/6·36-s + 1.31·37-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 66066 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.395572137\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.395572137\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 7 | \( 1 + T \) | |
| 11 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 2 T + p T^{2} \) | 1.41.c |
| 43 | \( 1 + 2 T + p T^{2} \) | 1.43.c |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 - 14 T + p T^{2} \) | 1.89.ao |
| 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
−14.46614111861449, −13.51282310559233, −13.19177968543203, −12.62598940807029, −11.92891770526975, −11.80191755692532, −11.04859758532826, −10.71425902561604, −10.02210657027958, −9.669534352854790, −9.326810141719836, −8.365186817621997, −8.163090336153529, −7.516880320770947, −6.871896192603639, −6.421286391129961, −5.955558602907601, −5.290773397711983, −4.759907595387762, −3.950257497698433, −3.317080039411551, −2.717022533018580, −1.886298446916746, −1.090447068402758, −0.5411001914602597,
0.5411001914602597, 1.090447068402758, 1.886298446916746, 2.717022533018580, 3.317080039411551, 3.950257497698433, 4.759907595387762, 5.290773397711983, 5.955558602907601, 6.421286391129961, 6.871896192603639, 7.516880320770947, 8.163090336153529, 8.365186817621997, 9.326810141719836, 9.669534352854790, 10.02210657027958, 10.71425902561604, 11.04859758532826, 11.80191755692532, 11.92891770526975, 12.62598940807029, 13.19177968543203, 13.51282310559233, 14.46614111861449
