| L(s) = 1 | + 3-s + 5-s − 7-s − 2·9-s + 3·11-s − 6·13-s + 15-s − 21-s − 2·23-s + 25-s − 5·27-s − 6·29-s + 3·33-s − 35-s − 37-s − 6·39-s − 9·41-s + 10·43-s − 2·45-s − 47-s − 6·49-s + 53-s + 3·55-s − 12·61-s + 2·63-s − 6·65-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 0.377·7-s − 2/3·9-s + 0.904·11-s − 1.66·13-s + 0.258·15-s − 0.218·21-s − 0.417·23-s + 1/5·25-s − 0.962·27-s − 1.11·29-s + 0.522·33-s − 0.169·35-s − 0.164·37-s − 0.960·39-s − 1.40·41-s + 1.52·43-s − 0.298·45-s − 0.145·47-s − 6/7·49-s + 0.137·53-s + 0.404·55-s − 1.53·61-s + 0.251·63-s − 0.744·65-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 - T \) | |
| 37 | \( 1 + T \) | |
| good | 3 | \( 1 - T + p T^{2} \) | 1.3.ab |
| 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 - 3 T + p T^{2} \) | 1.11.ad |
| 13 | \( 1 + 6 T + p T^{2} \) | 1.13.g |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + 2 T + p T^{2} \) | 1.23.c |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 - 10 T + p T^{2} \) | 1.43.ak |
| 47 | \( 1 + T + p T^{2} \) | 1.47.b |
| 53 | \( 1 - T + p T^{2} \) | 1.53.ab |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + 12 T + p T^{2} \) | 1.61.m |
| 67 | \( 1 + p T^{2} \) | 1.67.a |
| 71 | \( 1 - 5 T + p T^{2} \) | 1.71.af |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 11 T + p T^{2} \) | 1.83.l |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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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
−8.454669908512517207034899469370, −7.60623153841211391363563986699, −6.94299637524333632172982066183, −6.06920600436887629708345648593, −5.35180875187622235918959951272, −4.38768963429872255146818666900, −3.41377076281611111322968390654, −2.60923523958215693481233001490, −1.74532098704194312687928737335, 0,
1.74532098704194312687928737335, 2.60923523958215693481233001490, 3.41377076281611111322968390654, 4.38768963429872255146818666900, 5.35180875187622235918959951272, 6.06920600436887629708345648593, 6.94299637524333632172982066183, 7.60623153841211391363563986699, 8.454669908512517207034899469370
