| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 9-s − 11-s − 2·12-s + 13-s + 16-s − 3·17-s − 18-s + 4·19-s + 22-s + 6·23-s + 2·24-s − 26-s + 4·27-s − 3·29-s + 4·31-s − 32-s + 2·33-s + 3·34-s + 36-s + 10·37-s − 4·38-s − 2·39-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 1/3·9-s − 0.301·11-s − 0.577·12-s + 0.277·13-s + 1/4·16-s − 0.727·17-s − 0.235·18-s + 0.917·19-s + 0.213·22-s + 1.25·23-s + 0.408·24-s − 0.196·26-s + 0.769·27-s − 0.557·29-s + 0.718·31-s − 0.176·32-s + 0.348·33-s + 0.514·34-s + 1/6·36-s + 1.64·37-s − 0.648·38-s − 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| 13 | \( 1 - T \) | |
| good | 3 | \( 1 + 2 T + p T^{2} \) | 1.3.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 - 10 T + p T^{2} \) | 1.37.ak |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 + 8 T + p T^{2} \) | 1.61.i |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 2 T + p T^{2} \) | 1.73.ac |
| 79 | \( 1 - 11 T + p T^{2} \) | 1.79.al |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 + 10 T + p T^{2} \) | 1.97.k |
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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
−12.70744213117690, −12.08459891412945, −11.79030444948489, −11.26030832362585, −10.97141252354229, −10.73421418412270, −10.01409980894958, −9.701112247421666, −9.132626228692281, −8.761972427933200, −8.185673947725405, −7.671106598041933, −7.264449104576181, −6.704437604802370, −6.231721741976560, −5.971084595772814, −5.299696178226021, −4.849599946577289, −4.505024880937507, −3.656263506859593, −2.978562478726560, −2.692473994795118, −1.782828275569501, −1.211365524250837, −0.6544399120035253, 0,
0.6544399120035253, 1.211365524250837, 1.782828275569501, 2.692473994795118, 2.978562478726560, 3.656263506859593, 4.505024880937507, 4.849599946577289, 5.299696178226021, 5.971084595772814, 6.231721741976560, 6.704437604802370, 7.264449104576181, 7.671106598041933, 8.185673947725405, 8.761972427933200, 9.132626228692281, 9.701112247421666, 10.01409980894958, 10.73421418412270, 10.97141252354229, 11.26030832362585, 11.79030444948489, 12.08459891412945, 12.70744213117690
