| L(s) = 1 | + 3-s + 3·7-s + 9-s + 2·11-s + 13-s + 4·17-s + 3·21-s − 4·23-s − 5·25-s + 27-s + 7·31-s + 2·33-s − 37-s + 39-s + 4·41-s + 43-s − 2·47-s + 2·49-s + 4·51-s + 12·53-s − 2·59-s − 61-s + 3·63-s + 13·67-s − 4·69-s + 10·71-s − 3·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.970·17-s + 0.654·21-s − 0.834·23-s − 25-s + 0.192·27-s + 1.25·31-s + 0.348·33-s − 0.164·37-s + 0.160·39-s + 0.624·41-s + 0.152·43-s − 0.291·47-s + 2/7·49-s + 0.560·51-s + 1.64·53-s − 0.260·59-s − 0.128·61-s + 0.377·63-s + 1.58·67-s − 0.481·69-s + 1.18·71-s − 0.351·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8664 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.493116232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.493116232\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 19 | \( 1 \) | |
| good | 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 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 - 4 T + p T^{2} \) | 1.17.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 7 T + p T^{2} \) | 1.31.ah |
| 37 | \( 1 + T + p T^{2} \) | 1.37.b |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 - T + p T^{2} \) | 1.43.ab |
| 47 | \( 1 + 2 T + p T^{2} \) | 1.47.c |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 + 2 T + p T^{2} \) | 1.59.c |
| 61 | \( 1 + T + p T^{2} \) | 1.61.b |
| 67 | \( 1 - 13 T + p T^{2} \) | 1.67.an |
| 71 | \( 1 - 10 T + p T^{2} \) | 1.71.ak |
| 73 | \( 1 + 3 T + p T^{2} \) | 1.73.d |
| 79 | \( 1 + 13 T + p T^{2} \) | 1.79.n |
| 83 | \( 1 + 8 T + p T^{2} \) | 1.83.i |
| 89 | \( 1 - 18 T + p T^{2} \) | 1.89.as |
| 97 | \( 1 - 2 T + p T^{2} \) | 1.97.ac |
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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
−7.902267088556976333967452082296, −7.28424619249593199832951762759, −6.38002033203819681313592240946, −5.70647241645473426488394705692, −4.92088678133944556923239813316, −4.13076848286805938191727211639, −3.61095120770752977395398445272, −2.54765611264509120880499664182, −1.76323454857306749702407743571, −0.948354041979722767786019274465,
0.948354041979722767786019274465, 1.76323454857306749702407743571, 2.54765611264509120880499664182, 3.61095120770752977395398445272, 4.13076848286805938191727211639, 4.92088678133944556923239813316, 5.70647241645473426488394705692, 6.38002033203819681313592240946, 7.28424619249593199832951762759, 7.902267088556976333967452082296
