| L(s) = 1 | − 3·7-s − 2·11-s + 13-s − 8·17-s − 5·19-s + 23-s − 5·31-s − 2·37-s − 7·43-s − 6·47-s + 2·49-s − 6·59-s − 7·61-s + 13·67-s + 16·71-s + 10·73-s + 6·77-s − 8·79-s + 6·83-s − 10·89-s − 3·91-s − 13·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 0.603·11-s + 0.277·13-s − 1.94·17-s − 1.14·19-s + 0.208·23-s − 0.898·31-s − 0.328·37-s − 1.06·43-s − 0.875·47-s + 2/7·49-s − 0.781·59-s − 0.896·61-s + 1.58·67-s + 1.89·71-s + 1.17·73-s + 0.683·77-s − 0.900·79-s + 0.658·83-s − 1.05·89-s − 0.314·91-s − 1.31·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) | |
| 23 | \( 1 - T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 8 T + p T^{2} \) | 1.17.i |
| 19 | \( 1 + 5 T + p T^{2} \) | 1.19.f |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 + 5 T + p T^{2} \) | 1.31.f |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 + p T^{2} \) | 1.41.a |
| 43 | \( 1 + 7 T + p T^{2} \) | 1.43.h |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 7 T + p T^{2} \) | 1.61.h |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 16 T + p T^{2} \) | 1.71.aq |
| 73 | \( 1 - 10 T + p T^{2} \) | 1.73.ak |
| 79 | \( 1 + 8 T + p T^{2} \) | 1.79.i |
| 83 | \( 1 - 6 T + p T^{2} \) | 1.83.ag |
| 89 | \( 1 + 10 T + p T^{2} \) | 1.89.k |
| 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
−15.40550935121004, −14.92186082803573, −14.12084884084374, −13.56053096119571, −13.16122012112374, −12.73048349844305, −12.40518397414446, −11.49628907948975, −10.92719994845480, −10.73962550359201, −9.971340400373489, −9.416875409677651, −9.009809203935833, −8.342217514414652, −7.945127890013382, −6.948704998957595, −6.600296223649484, −6.331525603945127, −5.377887570209285, −4.915755123183643, −4.062379245656922, −3.672011250224249, −2.779311664394887, −2.309417571934748, −1.486755817171570, 0, 0,
1.486755817171570, 2.309417571934748, 2.779311664394887, 3.672011250224249, 4.062379245656922, 4.915755123183643, 5.377887570209285, 6.331525603945127, 6.600296223649484, 6.948704998957595, 7.945127890013382, 8.342217514414652, 9.009809203935833, 9.416875409677651, 9.971340400373489, 10.73962550359201, 10.92719994845480, 11.49628907948975, 12.40518397414446, 12.73048349844305, 13.16122012112374, 13.56053096119571, 14.12084884084374, 14.92186082803573, 15.40550935121004
