| L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 11-s − 4·13-s − 3·14-s + 16-s − 7·17-s + 5·19-s − 22-s − 23-s − 4·26-s − 3·28-s + 10·29-s + 2·31-s + 32-s − 7·34-s + 7·37-s + 5·38-s + 3·41-s − 4·43-s − 44-s − 46-s + 13·47-s + 2·49-s − 4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 0.301·11-s − 1.10·13-s − 0.801·14-s + 1/4·16-s − 1.69·17-s + 1.14·19-s − 0.213·22-s − 0.208·23-s − 0.784·26-s − 0.566·28-s + 1.85·29-s + 0.359·31-s + 0.176·32-s − 1.20·34-s + 1.15·37-s + 0.811·38-s + 0.468·41-s − 0.609·43-s − 0.150·44-s − 0.147·46-s + 1.89·47-s + 2/7·49-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.248576143\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.248576143\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 5 | \( 1 \) | |
| 11 | \( 1 + T \) | |
| good | 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 4 T + p T^{2} \) | 1.13.e |
| 17 | \( 1 + 7 T + p T^{2} \) | 1.17.h |
| 19 | \( 1 - 5 T + p T^{2} \) | 1.19.af |
| 23 | \( 1 + T + p T^{2} \) | 1.23.b |
| 29 | \( 1 - 10 T + p T^{2} \) | 1.29.ak |
| 31 | \( 1 - 2 T + p T^{2} \) | 1.31.ac |
| 37 | \( 1 - 7 T + p T^{2} \) | 1.37.ah |
| 41 | \( 1 - 3 T + p T^{2} \) | 1.41.ad |
| 43 | \( 1 + 4 T + p T^{2} \) | 1.43.e |
| 47 | \( 1 - 13 T + p T^{2} \) | 1.47.an |
| 53 | \( 1 - 4 T + p T^{2} \) | 1.53.ae |
| 59 | \( 1 - 5 T + p T^{2} \) | 1.59.af |
| 61 | \( 1 - 12 T + p T^{2} \) | 1.61.am |
| 67 | \( 1 - 12 T + p T^{2} \) | 1.67.am |
| 71 | \( 1 - 3 T + p T^{2} \) | 1.71.ad |
| 73 | \( 1 + 4 T + p T^{2} \) | 1.73.e |
| 79 | \( 1 + 15 T + p T^{2} \) | 1.79.p |
| 83 | \( 1 - 14 T + p T^{2} \) | 1.83.ao |
| 89 | \( 1 + p T^{2} \) | 1.89.a |
| 97 | \( 1 - 7 T + p T^{2} \) | 1.97.ah |
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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
−8.170229744447873333144637232417, −7.24897868147530309863376863288, −6.76973702247208580143143413511, −6.12333945055648408974489211805, −5.25370407329784117428093366354, −4.56109795419089017749867773435, −3.80440376571353822541164685890, −2.72525903842582848013736564209, −2.42256845207146758924999595899, −0.70656445460318542479657385921,
0.70656445460318542479657385921, 2.42256845207146758924999595899, 2.72525903842582848013736564209, 3.80440376571353822541164685890, 4.56109795419089017749867773435, 5.25370407329784117428093366354, 6.12333945055648408974489211805, 6.76973702247208580143143413511, 7.24897868147530309863376863288, 8.170229744447873333144637232417
