| L(s) = 1 | + 3·7-s − 3·9-s + 2·11-s − 2·13-s − 3·17-s − 23-s − 5·25-s + 6·29-s + 4·31-s − 4·37-s − 7·41-s + 8·43-s + 2·49-s + 12·53-s − 14·59-s + 6·61-s − 9·63-s + 2·67-s − 9·71-s + 3·73-s + 6·77-s + 9·81-s − 2·83-s − 89-s − 6·91-s − 14·97-s − 6·99-s + ⋯ |
| L(s) = 1 | + 1.13·7-s − 9-s + 0.603·11-s − 0.554·13-s − 0.727·17-s − 0.208·23-s − 25-s + 1.11·29-s + 0.718·31-s − 0.657·37-s − 1.09·41-s + 1.21·43-s + 2/7·49-s + 1.64·53-s − 1.82·59-s + 0.768·61-s − 1.13·63-s + 0.244·67-s − 1.06·71-s + 0.351·73-s + 0.683·77-s + 81-s − 0.219·83-s − 0.105·89-s − 0.628·91-s − 1.42·97-s − 0.603·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.928648641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.928648641\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 79 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 5 | \( 1 + p T^{2} \) | 1.5.a |
| 7 | \( 1 - 3 T + p T^{2} \) | 1.7.ad |
| 11 | \( 1 - 2 T + p T^{2} \) | 1.11.ac |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 + 3 T + p T^{2} \) | 1.17.d |
| 19 | \( 1 + p T^{2} \) | 1.19.a |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 6 T + p T^{2} \) | 1.29.ag |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 4 T + p T^{2} \) | 1.37.e |
| 41 | \( 1 + 7 T + p T^{2} \) | 1.41.h |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + p T^{2} \) | 1.47.a |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 14 T + p T^{2} \) | 1.59.o |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 9 T + p T^{2} \) | 1.71.j |
| 73 | \( 1 - 3 T + p T^{2} \) | 1.73.ad |
| 83 | \( 1 + 2 T + p T^{2} \) | 1.83.c |
| 89 | \( 1 + T + p T^{2} \) | 1.89.b |
| 97 | \( 1 + 14 T + p T^{2} \) | 1.97.o |
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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
−13.85392983341274, −13.51070320049834, −12.68684490177863, −12.08430409878384, −11.72342974580183, −11.53131515421566, −10.79009155527951, −10.44620598440685, −9.800656215956206, −9.223561141614132, −8.602728394665634, −8.441784026999492, −7.769669618587392, −7.322806641170819, −6.575917059403855, −6.201706207842362, −5.473171922450024, −5.078709201760019, −4.415811097211762, −4.033498487738728, −3.185388779036137, −2.547394689229884, −2.009293537765588, −1.323288907071228, −0.4377693814875211,
0.4377693814875211, 1.323288907071228, 2.009293537765588, 2.547394689229884, 3.185388779036137, 4.033498487738728, 4.415811097211762, 5.078709201760019, 5.473171922450024, 6.201706207842362, 6.575917059403855, 7.322806641170819, 7.769669618587392, 8.441784026999492, 8.602728394665634, 9.223561141614132, 9.800656215956206, 10.44620598440685, 10.79009155527951, 11.53131515421566, 11.72342974580183, 12.08430409878384, 12.68684490177863, 13.51070320049834, 13.85392983341274
