| L(s) = 1 | − 3-s − 5-s + 9-s − 2·11-s + 6·13-s + 15-s + 8·17-s + 4·19-s − 4·23-s + 25-s − 27-s + 2·29-s + 4·31-s + 2·33-s − 2·37-s − 6·39-s + 4·41-s − 6·43-s − 45-s + 8·47-s − 8·51-s + 2·53-s + 2·55-s − 4·57-s − 4·59-s + 10·61-s − 6·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 1.66·13-s + 0.258·15-s + 1.94·17-s + 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s + 0.718·31-s + 0.348·33-s − 0.328·37-s − 0.960·39-s + 0.624·41-s − 0.914·43-s − 0.149·45-s + 1.16·47-s − 1.12·51-s + 0.274·53-s + 0.269·55-s − 0.529·57-s − 0.520·59-s + 1.28·61-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 94080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 94080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.462532007\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.462532007\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) | |
| 5 | \( 1 + T \) | |
| 7 | \( 1 \) | |
| good | 11 | \( 1 + 2 T + p T^{2} \) | 1.11.c |
| 13 | \( 1 - 6 T + p T^{2} \) | 1.13.ag |
| 17 | \( 1 - 8 T + p T^{2} \) | 1.17.ai |
| 19 | \( 1 - 4 T + p T^{2} \) | 1.19.ae |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 4 T + p T^{2} \) | 1.31.ae |
| 37 | \( 1 + 2 T + p T^{2} \) | 1.37.c |
| 41 | \( 1 - 4 T + p T^{2} \) | 1.41.ae |
| 43 | \( 1 + 6 T + p T^{2} \) | 1.43.g |
| 47 | \( 1 - 8 T + p T^{2} \) | 1.47.ai |
| 53 | \( 1 - 2 T + p T^{2} \) | 1.53.ac |
| 59 | \( 1 + 4 T + p T^{2} \) | 1.59.e |
| 61 | \( 1 - 10 T + p T^{2} \) | 1.61.ak |
| 67 | \( 1 - 2 T + p T^{2} \) | 1.67.ac |
| 71 | \( 1 + 8 T + p T^{2} \) | 1.71.i |
| 73 | \( 1 + 12 T + p T^{2} \) | 1.73.m |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 - 16 T + p T^{2} \) | 1.89.aq |
| 97 | \( 1 + 8 T + p T^{2} \) | 1.97.i |
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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
−13.74623181641141, −13.29418276821795, −12.88766536959379, −12.10081340921181, −11.84669258548263, −11.59781254863259, −10.68305540637161, −10.50509020959028, −10.02206385983729, −9.402842795700367, −8.774999387803953, −8.124877305127688, −7.876738275099366, −7.314872837596026, −6.676715206915648, −5.984228651297343, −5.701341063032120, −5.181304037463131, −4.475830319426476, −3.821199543340537, −3.367759766637862, −2.826001515287789, −1.794358024969759, −1.073546892977079, −0.6267068265942104,
0.6267068265942104, 1.073546892977079, 1.794358024969759, 2.826001515287789, 3.367759766637862, 3.821199543340537, 4.475830319426476, 5.181304037463131, 5.701341063032120, 5.984228651297343, 6.676715206915648, 7.314872837596026, 7.876738275099366, 8.124877305127688, 8.774999387803953, 9.402842795700367, 10.02206385983729, 10.50509020959028, 10.68305540637161, 11.59781254863259, 11.84669258548263, 12.10081340921181, 12.88766536959379, 13.29418276821795, 13.74623181641141
