| L(s) = 1 | + 2·5-s − 2·7-s − 3·9-s − 2·11-s − 13-s − 7·17-s + 19-s − 5·23-s − 25-s + 2·29-s − 3·31-s − 4·35-s + 7·37-s + 7·41-s − 9·43-s − 6·45-s − 2·47-s − 3·49-s + 6·53-s − 4·55-s + 3·59-s + 11·61-s + 6·63-s − 2·65-s − 3·67-s − 16·71-s + 2·73-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s − 9-s − 0.603·11-s − 0.277·13-s − 1.69·17-s + 0.229·19-s − 1.04·23-s − 1/5·25-s + 0.371·29-s − 0.538·31-s − 0.676·35-s + 1.15·37-s + 1.09·41-s − 1.37·43-s − 0.894·45-s − 0.291·47-s − 3/7·49-s + 0.824·53-s − 0.539·55-s + 0.390·59-s + 1.40·61-s + 0.755·63-s − 0.248·65-s − 0.366·67-s − 1.89·71-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 988 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 988 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| 19 | \( 1 - T \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 2 T + p T^{2} \) | 1.7.c |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 23 | \( 1 + 5 T + p T^{2} \) | 1.23.f |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 3 T + p T^{2} \) | 1.31.d |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 7 T + p T^{2} \) | 1.41.ah |
| 43 | \( 1 + 9 T + p T^{2} \) | 1.43.j |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 - 3 T + p T^{2} \) | 1.59.ad |
| 61 | \( 1 - 11 T + p T^{2} \) | 1.61.al |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 16 T + p T^{2} \) | 1.71.q |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 11 T + p T^{2} \) | 1.97.l |
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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
−9.618311670709962385928710045291, −8.852788584124571379023099594667, −8.017929270255846081027719057360, −6.84712659276725306895566889448, −6.08513441068412309841685958980, −5.43118747128809654374806358340, −4.24150873987783380110088283702, −2.89408369476180701304993257526, −2.13010013912102737199678255096, 0,
2.13010013912102737199678255096, 2.89408369476180701304993257526, 4.24150873987783380110088283702, 5.43118747128809654374806358340, 6.08513441068412309841685958980, 6.84712659276725306895566889448, 8.017929270255846081027719057360, 8.852788584124571379023099594667, 9.618311670709962385928710045291
