| L(s) = 1 | − 2-s + 4-s − 7-s − 8-s − 4·13-s + 14-s + 16-s − 3·17-s − 4·19-s + 4·26-s − 28-s + 6·29-s − 4·31-s − 32-s + 3·34-s − 4·37-s + 4·38-s + 9·41-s + 2·43-s + 3·47-s − 6·49-s − 4·52-s − 6·53-s + 56-s − 6·58-s + 14·61-s + 4·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 0.727·17-s − 0.917·19-s + 0.784·26-s − 0.188·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.514·34-s − 0.657·37-s + 0.648·38-s + 1.40·41-s + 0.304·43-s + 0.437·47-s − 6/7·49-s − 0.554·52-s − 0.824·53-s + 0.133·56-s − 0.787·58-s + 1.79·61-s + 0.508·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8734857048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8734857048\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 + T + p T^{2} \) | 1.7.b |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 - 9 T + p T^{2} \) | 1.41.aj |
| 43 | \( 1 - 2 T + p T^{2} \) | 1.43.ac |
| 47 | \( 1 - 3 T + p T^{2} \) | 1.47.ad |
| 53 | \( 1 + 6 T + p T^{2} \) | 1.53.g |
| 59 | \( 1 + p T^{2} \) | 1.59.a |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 + T + p T^{2} \) | 1.73.b |
| 79 | \( 1 + T + p T^{2} \) | 1.79.b |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 - 3 T + p T^{2} \) | 1.89.ad |
| 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
−8.398800577476243321833055962044, −7.85406146058898438905526063811, −6.87084655478927532882563686211, −6.58395717502770172007587102871, −5.53984831485071354347351361589, −4.69220297031318818745509140462, −3.78314451123949742010402053025, −2.66680823999025742598609368482, −2.02736023469198575451261993286, −0.56776884864590562648620890187,
0.56776884864590562648620890187, 2.02736023469198575451261993286, 2.66680823999025742598609368482, 3.78314451123949742010402053025, 4.69220297031318818745509140462, 5.53984831485071354347351361589, 6.58395717502770172007587102871, 6.87084655478927532882563686211, 7.85406146058898438905526063811, 8.398800577476243321833055962044
