| L(s) = 1 | + 6·11-s + 4·13-s + 19-s + 6·23-s − 5·25-s − 6·29-s − 4·31-s + 2·37-s − 6·41-s − 8·43-s − 12·47-s − 6·53-s + 12·59-s + 10·61-s − 14·67-s − 12·71-s − 2·73-s + 10·79-s − 12·83-s + 18·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | + 1.80·11-s + 1.10·13-s + 0.229·19-s + 1.25·23-s − 25-s − 1.11·29-s − 0.718·31-s + 0.328·37-s − 0.937·41-s − 1.21·43-s − 1.75·47-s − 0.824·53-s + 1.56·59-s + 1.28·61-s − 1.71·67-s − 1.42·71-s − 0.234·73-s + 1.12·79-s − 1.31·83-s + 1.90·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 134064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 134064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| 19 | \( 1 - T \) | |
| good | 5 | \( 1 + p T^{2} \) | 1.5.a |
| 11 | \( 1 - 6 T + p T^{2} \) | 1.11.ag |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 + p T^{2} \) | 1.17.a |
| 23 | \( 1 - 6 T + p T^{2} \) | 1.23.ag |
| 29 | \( 1 + 6 T + p T^{2} \) | 1.29.g |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 + 8 T + p T^{2} \) | 1.43.i |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 - 12 T + p T^{2} \) | 1.59.am |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 + 14 T + p T^{2} \) | 1.67.o |
| 71 | \( 1 + 12 T + p T^{2} \) | 1.71.m |
| 73 | \( 1 + 2 T + p T^{2} \) | 1.73.c |
| 79 | \( 1 - 10 T + p T^{2} \) | 1.79.ak |
| 83 | \( 1 + 12 T + p T^{2} \) | 1.83.m |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.52714376993621, −13.16810310293013, −13.02143099418106, −12.02323565829346, −11.72508817805789, −11.41668387293917, −10.92970384708117, −10.34550970181127, −9.620322501030031, −9.396480309389631, −8.862151771084740, −8.400126925656587, −7.899760419793441, −7.117295832095659, −6.779357173460880, −6.344750540513475, −5.703853049009603, −5.286903151386506, −4.478002427098633, −4.017306211378040, −3.369497330818449, −3.211693446992163, −1.953526569819023, −1.588710551080328, −1.003762910100083, 0,
1.003762910100083, 1.588710551080328, 1.953526569819023, 3.211693446992163, 3.369497330818449, 4.017306211378040, 4.478002427098633, 5.286903151386506, 5.703853049009603, 6.344750540513475, 6.779357173460880, 7.117295832095659, 7.899760419793441, 8.400126925656587, 8.862151771084740, 9.396480309389631, 9.620322501030031, 10.34550970181127, 10.92970384708117, 11.41668387293917, 11.72508817805789, 12.02323565829346, 13.02143099418106, 13.16810310293013, 13.52714376993621
