| L(s) = 1 | + 3-s + 2·5-s − 3·7-s + 9-s − 11-s − 2·13-s + 2·15-s − 17-s − 2·19-s − 3·21-s + 2·23-s − 25-s + 27-s − 9·29-s + 8·31-s − 33-s − 6·35-s + 12·37-s − 2·39-s − 3·41-s + 2·45-s + 5·47-s + 2·49-s − 51-s − 11·53-s − 2·55-s − 2·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.516·15-s − 0.242·17-s − 0.458·19-s − 0.654·21-s + 0.417·23-s − 1/5·25-s + 0.192·27-s − 1.67·29-s + 1.43·31-s − 0.174·33-s − 1.01·35-s + 1.97·37-s − 0.320·39-s − 0.468·41-s + 0.298·45-s + 0.729·47-s + 2/7·49-s − 0.140·51-s − 1.51·53-s − 0.269·55-s − 0.264·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.159964981\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159964981\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) | |
| 11 | \( 1 + T \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 - 2 T + p T^{2} \) | 1.5.ac |
| 7 | \( 1 + 3 T + p T^{2} \) | 1.7.d |
| 13 | \( 1 + 2 T + p T^{2} \) | 1.13.c |
| 19 | \( 1 + 2 T + p T^{2} \) | 1.19.c |
| 23 | \( 1 - 2 T + p T^{2} \) | 1.23.ac |
| 29 | \( 1 + 9 T + p T^{2} \) | 1.29.j |
| 31 | \( 1 - 8 T + p T^{2} \) | 1.31.ai |
| 37 | \( 1 - 12 T + p T^{2} \) | 1.37.am |
| 41 | \( 1 + 3 T + p T^{2} \) | 1.41.d |
| 43 | \( 1 + p T^{2} \) | 1.43.a |
| 47 | \( 1 - 5 T + p T^{2} \) | 1.47.af |
| 53 | \( 1 + 11 T + p T^{2} \) | 1.53.l |
| 59 | \( 1 + 7 T + p T^{2} \) | 1.59.h |
| 61 | \( 1 - 2 T + p T^{2} \) | 1.61.ac |
| 67 | \( 1 + 3 T + p T^{2} \) | 1.67.d |
| 71 | \( 1 + 4 T + p T^{2} \) | 1.71.e |
| 73 | \( 1 + 13 T + p T^{2} \) | 1.73.n |
| 79 | \( 1 - 8 T + p T^{2} \) | 1.79.ai |
| 83 | \( 1 + 16 T + p T^{2} \) | 1.83.q |
| 89 | \( 1 - 15 T + p T^{2} \) | 1.89.ap |
| 97 | \( 1 - 16 T + p T^{2} \) | 1.97.aq |
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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
−14.94656756778478, −14.35869850338854, −13.77630552754563, −13.33534465721852, −12.79699264097657, −12.73478228168091, −11.72336678735815, −11.26149891873842, −10.37840370508940, −10.09825235118647, −9.506429521642068, −9.220063184594796, −8.622587269367510, −7.741507415867699, −7.463016771187249, −6.605413468146045, −6.147502514706394, −5.743112303619464, −4.799559535008497, −4.321923958818811, −3.420468269998941, −2.881382152528596, −2.305358835608659, −1.636275705155158, −0.5024622316025042,
0.5024622316025042, 1.636275705155158, 2.305358835608659, 2.881382152528596, 3.420468269998941, 4.321923958818811, 4.799559535008497, 5.743112303619464, 6.147502514706394, 6.605413468146045, 7.463016771187249, 7.741507415867699, 8.622587269367510, 9.220063184594796, 9.506429521642068, 10.09825235118647, 10.37840370508940, 11.26149891873842, 11.72336678735815, 12.73478228168091, 12.79699264097657, 13.33534465721852, 13.77630552754563, 14.35869850338854, 14.94656756778478
