| L(s) = 1 | − 3·9-s − 4·13-s − 4·17-s − 4·19-s + 8·23-s + 2·29-s + 8·31-s − 8·37-s − 6·41-s + 8·43-s − 8·47-s + 4·59-s + 6·61-s + 8·67-s + 12·71-s + 4·73-s − 4·79-s + 9·81-s + 10·89-s + 12·97-s + 18·101-s + 8·103-s + 8·107-s + 14·109-s − 16·113-s + 12·117-s + ⋯ |
| L(s) = 1 | − 9-s − 1.10·13-s − 0.970·17-s − 0.917·19-s + 1.66·23-s + 0.371·29-s + 1.43·31-s − 1.31·37-s − 0.937·41-s + 1.21·43-s − 1.16·47-s + 0.520·59-s + 0.768·61-s + 0.977·67-s + 1.42·71-s + 0.468·73-s − 0.450·79-s + 81-s + 1.05·89-s + 1.21·97-s + 1.79·101-s + 0.788·103-s + 0.773·107-s + 1.34·109-s − 1.50·113-s + 1.10·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.274009138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.274009138\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) | |
| 7 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} \) | 1.3.a |
| 11 | \( 1 + p T^{2} \) | 1.11.a |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 4 T + p T^{2} \) | 1.17.e |
| 19 | \( 1 + 4 T + p T^{2} \) | 1.19.e |
| 23 | \( 1 - 8 T + p T^{2} \) | 1.23.ai |
| 29 | \( 1 - 2 T + p T^{2} \) | 1.29.ac |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 + 8 T + p T^{2} \) | 1.37.i |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 8 T + p T^{2} \) | 1.43.ai |
| 47 | \( 1 + 8 T + p T^{2} \) | 1.47.i |
| 53 | \( 1 + p T^{2} \) | 1.53.a |
| 59 | \( 1 - 4 T + p T^{2} \) | 1.59.ae |
| 61 | \( 1 - 6 T + p T^{2} \) | 1.61.ag |
| 67 | \( 1 - 8 T + p T^{2} \) | 1.67.ai |
| 71 | \( 1 - 12 T + p T^{2} \) | 1.71.am |
| 73 | \( 1 - 4 T + p T^{2} \) | 1.73.ae |
| 79 | \( 1 + 4 T + p T^{2} \) | 1.79.e |
| 83 | \( 1 + p T^{2} \) | 1.83.a |
| 89 | \( 1 - 10 T + p T^{2} \) | 1.89.ak |
| 97 | \( 1 - 12 T + p T^{2} \) | 1.97.am |
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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
−8.505432969312621323078259469890, −7.51360102405484237472354732696, −6.72363268636683065066764403646, −6.27035911518646745626706190247, −5.03996501234127687817449665328, −4.87739467512684545227662349368, −3.67072791233902434365744470068, −2.75408917943536131028056346488, −2.12745245137009365413028530006, −0.59014905591076193569156252049,
0.59014905591076193569156252049, 2.12745245137009365413028530006, 2.75408917943536131028056346488, 3.67072791233902434365744470068, 4.87739467512684545227662349368, 5.03996501234127687817449665328, 6.27035911518646745626706190247, 6.72363268636683065066764403646, 7.51360102405484237472354732696, 8.505432969312621323078259469890
