L(s) = 1 | + 2-s − 3-s + 4-s − 0.656·5-s − 6-s + 4.00·7-s + 8-s + 9-s − 0.656·10-s − 11-s − 12-s − 0.713·13-s + 4.00·14-s + 0.656·15-s + 16-s − 6.79·17-s + 18-s − 2.11·19-s − 0.656·20-s − 4.00·21-s − 22-s − 5.33·23-s − 24-s − 4.56·25-s − 0.713·26-s − 27-s + 4.00·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.293·5-s − 0.408·6-s + 1.51·7-s + 0.353·8-s + 0.333·9-s − 0.207·10-s − 0.301·11-s − 0.288·12-s − 0.197·13-s + 1.07·14-s + 0.169·15-s + 0.250·16-s − 1.64·17-s + 0.235·18-s − 0.484·19-s − 0.146·20-s − 0.873·21-s − 0.213·22-s − 1.11·23-s − 0.204·24-s − 0.913·25-s − 0.139·26-s − 0.192·27-s + 0.756·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
good | 5 | \( 1 + 0.656T + 5T^{2} \) |
| 7 | \( 1 - 4.00T + 7T^{2} \) |
| 13 | \( 1 + 0.713T + 13T^{2} \) |
| 17 | \( 1 + 6.79T + 17T^{2} \) |
| 19 | \( 1 + 2.11T + 19T^{2} \) |
| 23 | \( 1 + 5.33T + 23T^{2} \) |
| 29 | \( 1 + 3.73T + 29T^{2} \) |
| 31 | \( 1 + 6.16T + 31T^{2} \) |
| 37 | \( 1 + 4.72T + 37T^{2} \) |
| 41 | \( 1 - 8.71T + 41T^{2} \) |
| 43 | \( 1 - 1.79T + 43T^{2} \) |
| 47 | \( 1 + 8.19T + 47T^{2} \) |
| 53 | \( 1 - 1.24T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 8.18T + 73T^{2} \) |
| 79 | \( 1 - 8.71T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 5.56T + 97T^{2} \) |
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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
−7.84149053181722045920433135360, −7.40956518071740262333676724832, −6.43987374853243827449932050009, −5.74382731709573004649684035180, −4.96661869380357420380921138733, −4.40120866847078279051432903483, −3.76524095095624779966625101928, −2.26033909355797893785173711367, −1.73411199290295858794871102990, 0,
1.73411199290295858794871102990, 2.26033909355797893785173711367, 3.76524095095624779966625101928, 4.40120866847078279051432903483, 4.96661869380357420380921138733, 5.74382731709573004649684035180, 6.43987374853243827449932050009, 7.40956518071740262333676724832, 7.84149053181722045920433135360