| L(s) = 1 | − 2-s + 4-s + 4·7-s − 8-s + 13-s − 4·14-s + 16-s − 3·17-s − 4·19-s − 26-s + 4·28-s − 9·29-s − 4·31-s − 32-s + 3·34-s + 37-s + 4·38-s − 6·41-s − 8·43-s − 12·47-s + 9·49-s + 52-s − 6·53-s − 4·56-s + 9·58-s − 61-s + 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.51·7-s − 0.353·8-s + 0.277·13-s − 1.06·14-s + 1/4·16-s − 0.727·17-s − 0.917·19-s − 0.196·26-s + 0.755·28-s − 1.67·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s + 0.164·37-s + 0.648·38-s − 0.937·41-s − 1.21·43-s − 1.75·47-s + 9/7·49-s + 0.138·52-s − 0.824·53-s − 0.534·56-s + 1.18·58-s − 0.128·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - T + p T^{2} \) | 1.37.ab |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 11 T + p T^{2} \) | 1.73.l |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 97 | \( 1 + 2 T + p T^{2} \) | 1.97.c |
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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
−8.322261352428122815522282193834, −7.46512509378029236519993243948, −6.80087579601397684362177011732, −5.91041114062864876677840677493, −5.06952779734665021477855716462, −4.34465294045856962440555226992, −3.33816650738335651198890182055, −2.00733431640776429456940895581, −1.60351853611858337130726797819, 0,
1.60351853611858337130726797819, 2.00733431640776429456940895581, 3.33816650738335651198890182055, 4.34465294045856962440555226992, 5.06952779734665021477855716462, 5.91041114062864876677840677493, 6.80087579601397684362177011732, 7.46512509378029236519993243948, 8.322261352428122815522282193834
