| L(s) = 1 | + 3-s + 5-s + 9-s + 2·11-s + 13-s + 15-s − 4·17-s + 19-s − 4·23-s + 25-s + 27-s + 5·31-s + 2·33-s − 5·37-s + 39-s + 2·41-s + 9·43-s + 45-s + 2·47-s − 4·51-s + 12·53-s + 2·55-s + 57-s + 8·59-s − 14·61-s + 65-s − 9·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.603·11-s + 0.277·13-s + 0.258·15-s − 0.970·17-s + 0.229·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.898·31-s + 0.348·33-s − 0.821·37-s + 0.160·39-s + 0.312·41-s + 1.37·43-s + 0.149·45-s + 0.291·47-s − 0.560·51-s + 1.64·53-s + 0.269·55-s + 0.132·57-s + 1.04·59-s − 1.79·61-s + 0.124·65-s − 1.09·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.109981424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.109981424\) |
| \(L(\frac{3}{2})\) |
|
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.ac |
| 13 | \( 1 - T + p T^{2} \) | 1.13.ab |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 - T + p T^{2} \) | 1.19.ab |
| 23 | \( 1 + 4 T + p T^{2} \) | 1.23.e |
| 29 | \( 1 + p T^{2} \) | 1.29.a |
| 31 | \( 1 - 5 T + p T^{2} \) | 1.31.af |
| 37 | \( 1 + 5 T + p T^{2} \) | 1.37.f |
| 41 | \( 1 - 2 T + p T^{2} \) | 1.41.ac |
| 43 | \( 1 - 9 T + p T^{2} \) | 1.43.aj |
| 47 | \( 1 - 2 T + p T^{2} \) | 1.47.ac |
| 53 | \( 1 - 12 T + p T^{2} \) | 1.53.am |
| 59 | \( 1 - 8 T + p T^{2} \) | 1.59.ai |
| 61 | \( 1 + 14 T + p T^{2} \) | 1.61.o |
| 67 | \( 1 + 9 T + p T^{2} \) | 1.67.j |
| 71 | \( 1 + 2 T + p T^{2} \) | 1.71.c |
| 73 | \( 1 - T + p T^{2} \) | 1.73.ab |
| 79 | \( 1 - 3 T + p T^{2} \) | 1.79.ad |
| 83 | \( 1 - 18 T + p T^{2} \) | 1.83.as |
| 89 | \( 1 + 4 T + p T^{2} \) | 1.89.e |
| 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
−16.17342966698184, −15.94513784711159, −15.16930489463020, −14.72477447512251, −14.01854889587517, −13.60579298899106, −13.25109346729181, −12.27749143071003, −12.01425336144381, −11.15550374592128, −10.52821594940082, −10.02314164018630, −9.195389759081784, −8.949967821436554, −8.242322109157635, −7.521408727500023, −6.867604520671214, −6.214882798364837, −5.643821027825035, −4.639073820229124, −4.114691589524375, −3.330138220755551, −2.450021426604430, −1.825639985289201, −0.7928058856277476,
0.7928058856277476, 1.825639985289201, 2.450021426604430, 3.330138220755551, 4.114691589524375, 4.639073820229124, 5.643821027825035, 6.214882798364837, 6.867604520671214, 7.521408727500023, 8.242322109157635, 8.949967821436554, 9.195389759081784, 10.02314164018630, 10.52821594940082, 11.15550374592128, 12.01425336144381, 12.27749143071003, 13.25109346729181, 13.60579298899106, 14.01854889587517, 14.72477447512251, 15.16930489463020, 15.94513784711159, 16.17342966698184
