| L(s) = 1 | + 2·7-s + 4·11-s + 2·13-s + 6·17-s + 2·19-s − 23-s + 4·29-s + 8·31-s − 2·37-s + 6·43-s − 4·47-s − 3·49-s + 10·53-s − 4·59-s − 6·61-s + 10·67-s + 8·71-s + 10·73-s + 8·77-s − 14·79-s − 4·83-s + 6·89-s + 4·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.755·7-s + 1.20·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s − 0.208·23-s + 0.742·29-s + 1.43·31-s − 0.328·37-s + 0.914·43-s − 0.583·47-s − 3/7·49-s + 1.37·53-s − 0.520·59-s − 0.768·61-s + 1.22·67-s + 0.949·71-s + 1.17·73-s + 0.911·77-s − 1.57·79-s − 0.439·83-s + 0.635·89-s + 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.607058247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.607058247\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 23 | \( 1 + T \) | |
| good | 7 | \( 1 - 2 T + p T^{2} \) | 1.7.ac |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 - 2 T + p T^{2} \) | 1.13.ac |
| 17 | \( 1 - 6 T + p T^{2} \) | 1.17.ag |
| 19 | \( 1 - 2 T + p T^{2} \) | 1.19.ac |
| 29 | \( 1 - 4 T + p T^{2} \) | 1.29.ae |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 4 T + p T^{2} \) | 1.47.e |
| 53 | \( 1 - 10 T + p T^{2} \) | 1.53.ak |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 + 6 T + p T^{2} \) | 1.61.g |
| 67 | \( 1 - 10 T + p T^{2} \) | 1.67.ak |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 14 T + p T^{2} \) | 1.79.o |
| 83 | \( 1 + 4 T + p T^{2} \) | 1.83.e |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−15.68515522717372, −14.97850008500719, −14.44238980174833, −13.99201931158862, −13.74309420453206, −12.77638161736547, −12.21681968926568, −11.78044012763172, −11.35439496228335, −10.67665978765175, −9.972830177386924, −9.621956462350239, −8.776700477656931, −8.351296397387462, −7.767430167323621, −7.142444292806251, −6.380376523326560, −5.933238952682620, −5.134980967296261, −4.564018399774466, −3.790863724735110, −3.260537767214790, −2.313807618936620, −1.316071251873063, −0.9291562749025988,
0.9291562749025988, 1.316071251873063, 2.313807618936620, 3.260537767214790, 3.790863724735110, 4.564018399774466, 5.134980967296261, 5.933238952682620, 6.380376523326560, 7.142444292806251, 7.767430167323621, 8.351296397387462, 8.776700477656931, 9.621956462350239, 9.972830177386924, 10.67665978765175, 11.35439496228335, 11.78044012763172, 12.21681968926568, 12.77638161736547, 13.74309420453206, 13.99201931158862, 14.44238980174833, 14.97850008500719, 15.68515522717372
