| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s − 3·7-s + 8-s + 9-s − 5·11-s + 12-s − 4·13-s − 3·14-s + 16-s + 17-s + 18-s − 19-s − 3·21-s − 5·22-s + 4·23-s + 24-s − 4·26-s + 27-s − 3·28-s − 4·29-s − 31-s + 32-s − 5·33-s + 34-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 − 1.50·11-s + 0.288·12-s − 1.10·13-s − 0.801·14-s + 1/4·16-s + 0.242·17-s + 0.235·18-s − 0.229·19-s − 0.654·21-s − 1.06·22-s + 0.834·23-s + 0.204·24-s − 0.784·26-s + 0.192·27-s − 0.566·28-s − 0.742·29-s − 0.179·31-s + 0.176·32-s − 0.870·33-s + 0.171·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 17 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 19 | \( 1 + T + p T^{2} \) | 1.19.b |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 4 T + p T^{2} \) | 1.29.e |
| 31 | \( 1 + T + p T^{2} \) | 1.31.b |
| 37 | \( 1 - 9 T + p T^{2} \) | 1.37.aj |
| 41 | \( 1 + 10 T + p T^{2} \) | 1.41.k |
| 43 | \( 1 + 11 T + p T^{2} \) | 1.43.l |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 3 T + p T^{2} \) | 1.53.d |
| 59 | \( 1 + 8 T + p T^{2} \) | 1.59.i |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 - 7 T + p T^{2} \) | 1.67.ah |
| 71 | \( 1 - 14 T + p T^{2} \) | 1.71.ao |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 + 5 T + p T^{2} \) | 1.79.f |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 2 T + p T^{2} \) | 1.89.c |
| 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
−8.352929112961728280899734692116, −7.66445519535649845552736365187, −6.98206144750941278763521419807, −6.20687552459893569178903967153, −5.20192828523326710084652496156, −4.65174741569572174849678402560, −3.32293455941357984088037082894, −2.97338740971163441188841279163, −1.97760349001575332689705434741, 0,
1.97760349001575332689705434741, 2.97338740971163441188841279163, 3.32293455941357984088037082894, 4.65174741569572174849678402560, 5.20192828523326710084652496156, 6.20687552459893569178903967153, 6.98206144750941278763521419807, 7.66445519535649845552736365187, 8.352929112961728280899734692116
