| L(s) = 1 | − 3-s + 7-s − 2·9-s − 2·11-s − 3·13-s − 17-s − 7·19-s − 21-s + 5·27-s − 9·29-s + 9·31-s + 2·33-s + 8·37-s + 3·39-s − 4·41-s + 12·43-s + 9·47-s + 49-s + 51-s + 3·53-s + 7·57-s − 3·59-s + 5·61-s − 2·63-s + 2·67-s + 9·71-s + 15·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s − 2/3·9-s − 0.603·11-s − 0.832·13-s − 0.242·17-s − 1.60·19-s − 0.218·21-s + 0.962·27-s − 1.67·29-s + 1.61·31-s + 0.348·33-s + 1.31·37-s + 0.480·39-s − 0.624·41-s + 1.82·43-s + 1.31·47-s + 1/7·49-s + 0.140·51-s + 0.412·53-s + 0.927·57-s − 0.390·59-s + 0.640·61-s − 0.251·63-s + 0.244·67-s + 1.06·71-s + 1.75·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.319178047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.319178047\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 5 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 17 | \( 1 + T \) | |
| good | 3 | \( 1 + T + p T^{2} \) | 1.3.b |
| 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 19 | \( 1 + 7 T + p T^{2} \) | 1.19.h |
| 23 | \( 1 + p T^{2} \) | 1.23.a |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 9 T + p T^{2} \) | 1.31.aj |
| 37 | \( 1 - 8 T + p T^{2} \) | 1.37.ai |
| 41 | \( 1 + 4 T + p T^{2} \) | 1.41.e |
| 43 | \( 1 - 12 T + p T^{2} \) | 1.43.am |
| 47 | \( 1 - 9 T + p T^{2} \) | 1.47.aj |
| 53 | \( 1 - 3 T + p T^{2} \) | 1.53.ad |
| 59 | \( 1 + 3 T + p T^{2} \) | 1.59.d |
| 61 | \( 1 - 5 T + p T^{2} \) | 1.61.af |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 - 9 T + p T^{2} \) | 1.71.aj |
| 73 | \( 1 - 15 T + p T^{2} \) | 1.73.ap |
| 79 | \( 1 - 16 T + p T^{2} \) | 1.79.aq |
| 83 | \( 1 - 4 T + p T^{2} \) | 1.83.ae |
| 89 | \( 1 + 5 T + p T^{2} \) | 1.89.f |
| 97 | \( 1 - 5 T + p T^{2} \) | 1.97.af |
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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
−13.10565110775115, −12.46256144747812, −12.25173956322295, −11.70792286110803, −11.09132468916062, −10.81498133662991, −10.55935764600329, −9.715350648456613, −9.454439460336313, −8.730878251078015, −8.374875682293543, −7.754582632827338, −7.509194341507454, −6.638406957291550, −6.358373404424529, −5.780122931806186, −5.250837452706749, −4.887580526197522, −4.203975300337514, −3.854048939022778, −2.864514478266025, −2.336204552988641, −2.121788627645834, −0.9159663665040125, −0.4080142257835023,
0.4080142257835023, 0.9159663665040125, 2.121788627645834, 2.336204552988641, 2.864514478266025, 3.854048939022778, 4.203975300337514, 4.887580526197522, 5.250837452706749, 5.780122931806186, 6.358373404424529, 6.638406957291550, 7.509194341507454, 7.754582632827338, 8.374875682293543, 8.730878251078015, 9.454439460336313, 9.715350648456613, 10.55935764600329, 10.81498133662991, 11.09132468916062, 11.70792286110803, 12.25173956322295, 12.46256144747812, 13.10565110775115
