| L(s) = 1 | + 4·7-s − 4·11-s + 13-s − 4·17-s − 6·19-s − 5·25-s + 6·29-s − 4·31-s + 6·37-s + 6·41-s + 4·43-s + 8·47-s + 9·49-s − 6·53-s − 12·59-s + 10·61-s + 2·67-s + 12·71-s − 10·73-s − 16·77-s + 10·79-s − 4·83-s − 10·89-s + 4·91-s + 14·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 1.20·11-s + 0.277·13-s − 0.970·17-s − 1.37·19-s − 25-s + 1.11·29-s − 0.718·31-s + 0.986·37-s + 0.937·41-s + 0.609·43-s + 1.16·47-s + 9/7·49-s − 0.824·53-s − 1.56·59-s + 1.28·61-s + 0.244·67-s + 1.42·71-s − 1.17·73-s − 1.82·77-s + 1.12·79-s − 0.439·83-s − 1.05·89-s + 0.419·91-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 13 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 6 T + p T^{2} \) | 1.19.g |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 4 T + p T^{2} \) | 1.43.ae |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−16.16149798823752, −15.74161910949492, −15.21785180482846, −14.73114717642115, −14.06439712299145, −13.67409429377346, −12.88983840110262, −12.57254948030845, −11.71440257155431, −11.09678659177245, −10.82760033961999, −10.30887822000224, −9.393174261313591, −8.779334628918757, −8.109183262089287, −7.881829960588877, −7.131727040273547, −6.251824616231789, −5.715568826454950, −4.892724899406947, −4.466936763134407, −3.821384395000940, −2.477041311642609, −2.254841525012367, −1.206899576857504, 0,
1.206899576857504, 2.254841525012367, 2.477041311642609, 3.821384395000940, 4.466936763134407, 4.892724899406947, 5.715568826454950, 6.251824616231789, 7.131727040273547, 7.881829960588877, 8.109183262089287, 8.779334628918757, 9.393174261313591, 10.30887822000224, 10.82760033961999, 11.09678659177245, 11.71440257155431, 12.57254948030845, 12.88983840110262, 13.67409429377346, 14.06439712299145, 14.73114717642115, 15.21785180482846, 15.74161910949492, 16.16149798823752
