L(s) = 1 | − 3·7-s − 4·11-s + 3·17-s + 8·19-s + 2·29-s − 4·31-s + 8·37-s − 9·41-s − 6·43-s + 7·47-s + 2·49-s + 6·53-s − 12·59-s + 10·61-s + 10·67-s + 4·71-s − 9·73-s + 12·77-s + 5·79-s − 6·83-s + 13·89-s − 7·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | − 1.13·7-s − 1.20·11-s + 0.727·17-s + 1.83·19-s + 0.371·29-s − 0.718·31-s + 1.31·37-s − 1.40·41-s − 0.914·43-s + 1.02·47-s + 2/7·49-s + 0.824·53-s − 1.56·59-s + 1.28·61-s + 1.22·67-s + 0.474·71-s − 1.05·73-s + 1.36·77-s + 0.562·79-s − 0.658·83-s + 1.37·89-s − 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 5 | \( 1 \) | |
good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 4 T + p T^{2} \) | 1.11.e |
| 13 | \( 1 + p T^{2} \) | 1.13.a |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 9 T + p T^{2} \) | 1.41.j |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 7 T + p T^{2} \) | 1.47.ah |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 12 T + p T^{2} \) | 1.59.m |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 4 T + p T^{2} \) | 1.71.ae |
| 73 | \( 1 + 9 T + p T^{2} \) | 1.73.j |
| 79 | \( 1 - 5 T + p T^{2} \) | 1.79.af |
| 83 | \( 1 + 6 T + p T^{2} \) | 1.83.g |
| 89 | \( 1 - 13 T + p T^{2} \) | 1.89.an |
| 97 | \( 1 + 7 T + p T^{2} \) | 1.97.h |
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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
−13.64762988335868, −13.22995919987784, −12.86999035879428, −12.25097415656973, −11.93943177208836, −11.33294226293665, −10.80341022370489, −10.17116355458595, −9.858122968684781, −9.569581087458207, −8.877888831525185, −8.343086754735106, −7.622812330905188, −7.503398878631471, −6.796917271446398, −6.268732938654333, −5.654842081444044, −5.226031029116703, −4.839687033536509, −3.792423516918658, −3.502864504796305, −2.830941195638654, −2.503399005547663, −1.482821136251700, −0.7737478840733393, 0,
0.7737478840733393, 1.482821136251700, 2.503399005547663, 2.830941195638654, 3.502864504796305, 3.792423516918658, 4.839687033536509, 5.226031029116703, 5.654842081444044, 6.268732938654333, 6.796917271446398, 7.503398878631471, 7.622812330905188, 8.343086754735106, 8.877888831525185, 9.569581087458207, 9.858122968684781, 10.17116355458595, 10.80341022370489, 11.33294226293665, 11.93943177208836, 12.25097415656973, 12.86999035879428, 13.22995919987784, 13.64762988335868