| L(s) = 1 | − 3·7-s + 5·11-s + 4·13-s + 3·17-s − 4·19-s + 4·23-s − 3·29-s − 7·31-s + 37-s + 9·41-s + 5·43-s + 12·47-s + 2·49-s + 7·53-s + 8·59-s + 3·61-s − 8·67-s + 6·71-s − 10·73-s − 15·77-s + 8·79-s + 12·83-s + 12·89-s − 12·91-s − 13·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.13·7-s + 1.50·11-s + 1.10·13-s + 0.727·17-s − 0.917·19-s + 0.834·23-s − 0.557·29-s − 1.25·31-s + 0.164·37-s + 1.40·41-s + 0.762·43-s + 1.75·47-s + 2/7·49-s + 0.961·53-s + 1.04·59-s + 0.384·61-s − 0.977·67-s + 0.712·71-s − 1.17·73-s − 1.70·77-s + 0.900·79-s + 1.31·83-s + 1.27·89-s − 1.25·91-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 266400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 266400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 37 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 - 5 T + p T^{2} \) | 1.11.af |
| 13 | \( 1 - 4 T + p T^{2} \) | 1.13.ae |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + 3 T + p T^{2} \) | 1.29.d |
| 31 | \( 1 + 7 T + p T^{2} \) | 1.31.h |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 - 12 T + p T^{2} \) | 1.47.am |
| 53 | \( 1 - 7 T + p T^{2} \) | 1.53.ah |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 - 3 T + p T^{2} \) | 1.61.ad |
| 67 | \( 1 + 8 T + p T^{2} \) | 1.67.i |
| 71 | \( 1 - 6 T + p T^{2} \) | 1.71.ag |
| 73 | \( 1 + 10 T + p T^{2} \) | 1.73.k |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 12 T + p T^{2} \) | 1.89.am |
| 97 | \( 1 + 13 T + p T^{2} \) | 1.97.n |
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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
−13.00624808913720, −12.62156339471994, −12.16035790089624, −11.66198397070863, −11.17819938642899, −10.63675445494582, −10.43282587836621, −9.615684129424357, −9.217533403042715, −8.998989039380431, −8.603669924706068, −7.754856555221065, −7.388135868715974, −6.757620145702976, −6.470487043735775, −5.848676423488366, −5.700195060083493, −4.853126757733888, −4.020844392714566, −3.805943720886557, −3.511892249546908, −2.639941016250958, −2.186689525429908, −1.177192631702949, −0.9943438542850082, 0,
0.9943438542850082, 1.177192631702949, 2.186689525429908, 2.639941016250958, 3.511892249546908, 3.805943720886557, 4.020844392714566, 4.853126757733888, 5.700195060083493, 5.848676423488366, 6.470487043735775, 6.757620145702976, 7.388135868715974, 7.754856555221065, 8.603669924706068, 8.998989039380431, 9.217533403042715, 9.615684129424357, 10.43282587836621, 10.63675445494582, 11.17819938642899, 11.66198397070863, 12.16035790089624, 12.62156339471994, 13.00624808913720
