| L(s) = 1 | − 2-s + 4-s − 8-s − 5·11-s − 13-s + 16-s + 3·17-s + 5·19-s + 5·22-s + 9·23-s + 26-s + 29-s − 2·31-s − 32-s − 3·34-s − 3·37-s − 5·38-s + 12·41-s − 7·43-s − 5·44-s − 9·46-s − 4·47-s − 52-s − 10·53-s − 58-s − 14·59-s − 11·61-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 1.50·11-s − 0.277·13-s + 1/4·16-s + 0.727·17-s + 1.14·19-s + 1.06·22-s + 1.87·23-s + 0.196·26-s + 0.185·29-s − 0.359·31-s − 0.176·32-s − 0.514·34-s − 0.493·37-s − 0.811·38-s + 1.87·41-s − 1.06·43-s − 0.753·44-s − 1.32·46-s − 0.583·47-s − 0.138·52-s − 1.37·53-s − 0.131·58-s − 1.82·59-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 13 | \( 1 + T \) | |
| good | 11 | \( 1 + 5 T + p T^{2} \) | 1.11.f |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 - 9 T + p T^{2} \) | 1.23.aj |
| 29 | \( 1 - T + p T^{2} \) | 1.29.ab |
| 31 | \( 1 + 2 T + p T^{2} \) | 1.31.c |
| 37 | \( 1 + 3 T + p T^{2} \) | 1.37.d |
| 41 | \( 1 - 12 T + p T^{2} \) | 1.41.am |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 + 4 T + p T^{2} \) | 1.67.e |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 - 11 T + p T^{2} \) | 1.73.al |
| 79 | \( 1 - 4 T + p T^{2} \) | 1.79.ae |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 16 T + p T^{2} \) | 1.89.q |
| 97 | \( 1 - 14 T + p T^{2} \) | 1.97.ao |
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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
−12.75020237938623, −12.53124091908648, −12.12754869490660, −11.24258297772045, −11.17688663712806, −10.64590598669620, −10.21548806509863, −9.631394060379460, −9.361524591846732, −8.896734223079030, −8.202571420324750, −7.780859148055820, −7.542159869518496, −7.058534759622149, −6.439397759752817, −5.883955910542092, −5.301152201976362, −4.973387245905121, −4.525133238504550, −3.409117039596949, −3.160258837474673, −2.733707473451688, −2.004064398376170, −1.329024556356529, −0.7429540225462538, 0,
0.7429540225462538, 1.329024556356529, 2.004064398376170, 2.733707473451688, 3.160258837474673, 3.409117039596949, 4.525133238504550, 4.973387245905121, 5.301152201976362, 5.883955910542092, 6.439397759752817, 7.058534759622149, 7.542159869518496, 7.780859148055820, 8.202571420324750, 8.896734223079030, 9.361524591846732, 9.631394060379460, 10.21548806509863, 10.64590598669620, 11.17688663712806, 11.24258297772045, 12.12754869490660, 12.53124091908648, 12.75020237938623
