| L(s) = 1 | + 3·3-s + 5-s + 6·9-s − 11-s − 3·13-s + 3·15-s + 3·17-s + 6·19-s + 4·23-s + 25-s + 9·27-s − 29-s + 6·31-s − 3·33-s − 9·39-s − 6·41-s + 6·43-s + 6·45-s − 9·47-s + 9·51-s − 10·53-s − 55-s + 18·57-s − 6·59-s − 3·65-s + 14·67-s + 12·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 2·9-s − 0.301·11-s − 0.832·13-s + 0.774·15-s + 0.727·17-s + 1.37·19-s + 0.834·23-s + 1/5·25-s + 1.73·27-s − 0.185·29-s + 1.07·31-s − 0.522·33-s − 1.44·39-s − 0.937·41-s + 0.914·43-s + 0.894·45-s − 1.31·47-s + 1.26·51-s − 1.37·53-s − 0.134·55-s + 2.38·57-s − 0.781·59-s − 0.372·65-s + 1.71·67-s + 1.44·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.217053297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.217053297\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 - p T + p T^{2} \) | 1.3.ad |
| 11 | \( 1 + T + p T^{2} \) | 1.11.b |
| 13 | \( 1 + 3 T + p T^{2} \) | 1.13.d |
| 17 | \( 1 - 3 T + p T^{2} \) | 1.17.ad |
| 19 | \( 1 - 6 T + p T^{2} \) | 1.19.ag |
| 23 | \( 1 - 4 T + p T^{2} \) | 1.23.ae |
| 29 | \( 1 + T + p T^{2} \) | 1.29.b |
| 31 | \( 1 - 6 T + p T^{2} \) | 1.31.ag |
| 37 | \( 1 + p T^{2} \) | 1.37.a |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 6 T + p T^{2} \) | 1.43.ag |
| 47 | \( 1 + 9 T + p T^{2} \) | 1.47.j |
| 53 | \( 1 + 10 T + p T^{2} \) | 1.53.k |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + p T^{2} \) | 1.61.a |
| 67 | \( 1 - 14 T + p T^{2} \) | 1.67.ao |
| 71 | \( 1 - 8 T + p T^{2} \) | 1.71.ai |
| 73 | \( 1 + 6 T + p T^{2} \) | 1.73.g |
| 79 | \( 1 - T + p T^{2} \) | 1.79.ab |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 12 T + p T^{2} \) | 1.89.m |
| 97 | \( 1 - 15 T + p T^{2} \) | 1.97.ap |
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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
−8.413182972397047221081872457399, −7.80457391816395721714669401072, −7.30703478828022800288631659705, −6.46027048767020078983890297601, −5.27291442121198189570792311479, −4.68059577608072660097150741909, −3.46612148832537381816016895570, −3.00378692647977841566927386749, −2.19920226028059021829079345585, −1.18708333743445212285166420152,
1.18708333743445212285166420152, 2.19920226028059021829079345585, 3.00378692647977841566927386749, 3.46612148832537381816016895570, 4.68059577608072660097150741909, 5.27291442121198189570792311479, 6.46027048767020078983890297601, 7.30703478828022800288631659705, 7.80457391816395721714669401072, 8.413182972397047221081872457399
