| L(s) = 1 | − 2-s + 3-s − 4-s + 5-s − 6-s + 4·7-s + 3·8-s + 9-s − 10-s + 4·11-s − 12-s − 2·13-s − 4·14-s + 15-s − 16-s + 2·17-s − 18-s − 20-s + 4·21-s − 4·22-s − 4·23-s + 3·24-s + 25-s + 2·26-s + 27-s − 4·28-s + 2·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.51·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s − 0.554·13-s − 1.06·14-s + 0.258·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.223·20-s + 0.872·21-s − 0.852·22-s − 0.834·23-s + 0.612·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.755·28-s + 0.371·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5415 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.246838526\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.246838526\) |
| \(L(\frac{3}{2})\) |
|
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 | 3 | \( 1 - T \) | |
| 5 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 2 | \( 1 + T + p T^{2} \) | 1.2.b |
| 7 | \( 1 - 4 T + p T^{2} \) | 1.7.ae |
| 11 | \( 1 - 4 T + p T^{2} \) | 1.11.ae |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 17 | \( 1 - 2 T + p T^{2} \) | 1.17.ac |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 + p T^{2} \) | 1.31.a |
| 37 | \( 1 - 6 T + p T^{2} \) | 1.37.ag |
| 41 | \( 1 - 6 T + p T^{2} \) | 1.41.ag |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 12 T + p T^{2} \) | 1.47.m |
| 53 | \( 1 - 14 T + p T^{2} \) | 1.53.ao |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 14 T + p T^{2} \) | 1.61.ao |
| 67 | \( 1 - 4 T + p T^{2} \) | 1.67.ae |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 + 14 T + p T^{2} \) | 1.73.o |
| 79 | \( 1 + 16 T + p T^{2} \) | 1.79.q |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 6 T + p T^{2} \) | 1.89.ag |
| 97 | \( 1 - 10 T + p T^{2} \) | 1.97.ak |
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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
−8.341266231722576929355138064698, −7.66024161175802088465262717233, −7.10351864803638892904617636735, −5.99854802649507498936131353351, −5.17139269450503150803410429759, −4.38480439500327259414036248877, −3.92995143177201204644709512658, −2.53208153675291247558549077183, −1.65531003419966530014973287573, −0.983012618682254792025469910512,
0.983012618682254792025469910512, 1.65531003419966530014973287573, 2.53208153675291247558549077183, 3.92995143177201204644709512658, 4.38480439500327259414036248877, 5.17139269450503150803410429759, 5.99854802649507498936131353351, 7.10351864803638892904617636735, 7.66024161175802088465262717233, 8.341266231722576929355138064698
