| L(s) = 1 | + 2-s + 4-s − 4·7-s + 8-s − 11-s − 4·14-s + 16-s − 2·17-s + 8·19-s − 22-s + 4·23-s − 4·28-s − 8·31-s + 32-s − 2·34-s − 4·37-s + 8·38-s + 4·41-s − 8·43-s − 44-s + 4·46-s − 12·47-s + 9·49-s + 2·53-s − 4·56-s − 12·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.51·7-s + 0.353·8-s − 0.301·11-s − 1.06·14-s + 1/4·16-s − 0.485·17-s + 1.83·19-s − 0.213·22-s + 0.834·23-s − 0.755·28-s − 1.43·31-s + 0.176·32-s − 0.342·34-s − 0.657·37-s + 1.29·38-s + 0.624·41-s − 1.21·43-s − 0.150·44-s + 0.589·46-s − 1.75·47-s + 9/7·49-s + 0.274·53-s − 0.534·56-s − 1.56·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 4 T + p T^{2} \) | 1.7.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 + 2 T + p T^{2} \) | 1.17.c |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 8 T + p T^{2} \) | 1.31.i |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 + 16 T + p T^{2} \) | 1.97.q |
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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
−7.65503445966005832377121014513, −6.95357732142672684665681057726, −6.53668170022004700395203828967, −5.54317531553370379113419407133, −5.14397056712661196206888550988, −4.00790890036147446033281252921, −3.24113841118841835121981805379, −2.81738706083250408786678249491, −1.48564895312568376525360123883, 0,
1.48564895312568376525360123883, 2.81738706083250408786678249491, 3.24113841118841835121981805379, 4.00790890036147446033281252921, 5.14397056712661196206888550988, 5.54317531553370379113419407133, 6.53668170022004700395203828967, 6.95357732142672684665681057726, 7.65503445966005832377121014513
