| L(s) = 1 | − 3-s − 5-s + 3·7-s + 9-s + 5·13-s + 15-s − 4·17-s − 7·19-s − 3·21-s − 4·23-s + 25-s − 27-s + 3·29-s + 3·31-s − 3·35-s − 4·37-s − 5·39-s − 11·41-s + 43-s − 45-s − 2·47-s + 2·49-s + 4·51-s + 10·53-s + 7·57-s − 4·59-s − 11·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1.13·7-s + 1/3·9-s + 1.38·13-s + 0.258·15-s − 0.970·17-s − 1.60·19-s − 0.654·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.557·29-s + 0.538·31-s − 0.507·35-s − 0.657·37-s − 0.800·39-s − 1.71·41-s + 0.152·43-s − 0.149·45-s − 0.291·47-s + 2/7·49-s + 0.560·51-s + 1.37·53-s + 0.927·57-s − 0.520·59-s − 1.40·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 3 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 43 | \( 1 - T \) | |
| good | 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 - 5 T + p T^{2} \) | 1.13.af |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 3 T + p T^{2} \) | 1.29.ad |
| 31 | \( 1 - 3 T + p T^{2} \) | 1.31.ad |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 11 T + p T^{2} \) | 1.41.l |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 11 T + p T^{2} \) | 1.61.l |
| 67 | \( 1 - T + p T^{2} \) | 1.67.ab |
| 71 | \( 1 + 14 T + p T^{2} \) | 1.71.o |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 - 13 T + p T^{2} \) | 1.79.an |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 97 | \( 1 - 6 T + p T^{2} \) | 1.97.ag |
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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
−8.053554177900616896980213932582, −7.02832308323562366588618337748, −6.39327509507671339378389867715, −5.77422294721419503762185296234, −4.70192405105752995433631572599, −4.36785902876509993676743112370, −3.48705521644292636227371826959, −2.14839264109316259139552249880, −1.36799422266439006368547983451, 0,
1.36799422266439006368547983451, 2.14839264109316259139552249880, 3.48705521644292636227371826959, 4.36785902876509993676743112370, 4.70192405105752995433631572599, 5.77422294721419503762185296234, 6.39327509507671339378389867715, 7.02832308323562366588618337748, 8.053554177900616896980213932582
