| L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 8-s + 9-s + 2·11-s − 12-s − 3·13-s + 16-s − 17-s − 18-s + 2·19-s − 2·22-s − 2·23-s + 24-s + 3·26-s − 27-s + 8·29-s − 10·31-s − 32-s − 2·33-s + 34-s + 36-s − 12·37-s − 2·38-s + 3·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.603·11-s − 0.288·12-s − 0.832·13-s + 1/4·16-s − 0.242·17-s − 0.235·18-s + 0.458·19-s − 0.426·22-s − 0.417·23-s + 0.204·24-s + 0.588·26-s − 0.192·27-s + 1.48·29-s − 1.79·31-s − 0.176·32-s − 0.348·33-s + 0.171·34-s + 1/6·36-s − 1.97·37-s − 0.324·38-s + 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 - 8 T + p T^{2} \) | 1.29.ai |
| 31 | \( 1 + 10 T + p T^{2} \) | 1.31.k |
| 37 | \( 1 + 12 T + p T^{2} \) | 1.37.m |
| 41 | \( 1 - 10 T + p T^{2} \) | 1.41.ak |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 + 3 T + p T^{2} \) | 1.61.d |
| 67 | \( 1 - 9 T + p T^{2} \) | 1.67.aj |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 6 T + p T^{2} \) | 1.79.ag |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 17 T + p T^{2} \) | 1.97.ar |
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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
−13.81566499058857, −13.20977157611828, −12.56542409548795, −12.22961326195683, −11.77147106858269, −11.43310809637025, −10.72478876957337, −10.36147337439141, −9.994831582777652, −9.308441149358720, −8.991711562878129, −8.468136361337289, −7.798462718584817, −7.249525653630664, −6.950502910384521, −6.413043933484066, −5.755410316285015, −5.318648047450755, −4.725849448574083, −4.065218841697142, −3.484656298301802, −2.766972122336809, −2.057673744474393, −1.509261550500343, −0.7257814366571887, 0,
0.7257814366571887, 1.509261550500343, 2.057673744474393, 2.766972122336809, 3.484656298301802, 4.065218841697142, 4.725849448574083, 5.318648047450755, 5.755410316285015, 6.413043933484066, 6.950502910384521, 7.249525653630664, 7.798462718584817, 8.468136361337289, 8.991711562878129, 9.308441149358720, 9.994831582777652, 10.36147337439141, 10.72478876957337, 11.43310809637025, 11.77147106858269, 12.22961326195683, 12.56542409548795, 13.20977157611828, 13.81566499058857
