| L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s − 3·11-s + 2·15-s − 3·17-s + 19-s + 2·21-s + 4·23-s − 25-s − 27-s − 2·31-s + 3·33-s + 4·35-s − 10·37-s + 9·41-s − 2·45-s − 2·47-s − 3·49-s + 3·51-s + 12·53-s + 6·55-s − 57-s − 7·59-s + 4·61-s − 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.904·11-s + 0.516·15-s − 0.727·17-s + 0.229·19-s + 0.436·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 0.359·31-s + 0.522·33-s + 0.676·35-s − 1.64·37-s + 1.40·41-s − 0.298·45-s − 0.291·47-s − 3/7·49-s + 0.420·51-s + 1.64·53-s + 0.809·55-s − 0.132·57-s − 0.911·59-s + 0.512·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 355008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 43 | \( 1 \) | |
| good | 5 | \( 1 + 2 T + p T^{2} \) | 1.5.c |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 10 T + p T^{2} \) | 1.37.k |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 4 T + p T^{2} \) | 1.61.ae |
| 67 | \( 1 - 3 T + p T^{2} \) | 1.67.ad |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 + 9 T + p T^{2} \) | 1.89.j |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−12.75753061065313, −12.31300229458302, −11.85022451532334, −11.37867765529853, −10.97482703552713, −10.64167204333468, −10.05099403070562, −9.720121599074396, −9.091693573889532, −8.608468716048835, −8.252170681634965, −7.514573596678850, −7.206096759272852, −6.913052897163056, −6.252660929932883, −5.661998212070281, −5.408822403910457, −4.657305975241370, −4.334029258586779, −3.722632337113506, −3.200726351972375, −2.727483519640968, −2.053704286427562, −1.295912512201245, −0.4767936937703701, 0,
0.4767936937703701, 1.295912512201245, 2.053704286427562, 2.727483519640968, 3.200726351972375, 3.722632337113506, 4.334029258586779, 4.657305975241370, 5.408822403910457, 5.661998212070281, 6.252660929932883, 6.913052897163056, 7.206096759272852, 7.514573596678850, 8.252170681634965, 8.608468716048835, 9.091693573889532, 9.720121599074396, 10.05099403070562, 10.64167204333468, 10.97482703552713, 11.37867765529853, 11.85022451532334, 12.31300229458302, 12.75753061065313
