L(s) = 1 | − 2.24·2-s + 1.35·3-s + 3.04·4-s − 5-s − 3.04·6-s − 7-s − 2.35·8-s − 1.15·9-s + 2.24·10-s − 0.753·11-s + 4.13·12-s + 3.58·13-s + 2.24·14-s − 1.35·15-s − 0.801·16-s − 2.85·17-s + 2.60·18-s + 1.04·19-s − 3.04·20-s − 1.35·21-s + 1.69·22-s − 23-s − 3.19·24-s + 25-s − 8.04·26-s − 5.64·27-s − 3.04·28-s + ⋯ |
L(s) = 1 | − 1.58·2-s + 0.783·3-s + 1.52·4-s − 0.447·5-s − 1.24·6-s − 0.377·7-s − 0.833·8-s − 0.386·9-s + 0.710·10-s − 0.227·11-s + 1.19·12-s + 0.993·13-s + 0.600·14-s − 0.350·15-s − 0.200·16-s − 0.691·17-s + 0.613·18-s + 0.240·19-s − 0.681·20-s − 0.296·21-s + 0.360·22-s − 0.208·23-s − 0.652·24-s + 0.200·25-s − 1.57·26-s − 1.08·27-s − 0.576·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 2.24T + 2T^{2} \) |
| 3 | \( 1 - 1.35T + 3T^{2} \) |
| 11 | \( 1 + 0.753T + 11T^{2} \) |
| 13 | \( 1 - 3.58T + 13T^{2} \) |
| 17 | \( 1 + 2.85T + 17T^{2} \) |
| 19 | \( 1 - 1.04T + 19T^{2} \) |
| 29 | \( 1 + 7.38T + 29T^{2} \) |
| 31 | \( 1 + 1.44T + 31T^{2} \) |
| 37 | \( 1 + 3.74T + 37T^{2} \) |
| 41 | \( 1 + 8.70T + 41T^{2} \) |
| 43 | \( 1 - 2.24T + 43T^{2} \) |
| 47 | \( 1 + 1.76T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 + 2.59T + 59T^{2} \) |
| 61 | \( 1 + 8.76T + 61T^{2} \) |
| 67 | \( 1 - 6.47T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 - 2.76T + 73T^{2} \) |
| 79 | \( 1 + 11.0T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 - 3.46T + 89T^{2} \) |
| 97 | \( 1 + 0.560T + 97T^{2} \) |
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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
−9.562417468251677658992029589314, −8.842074896028910377868684174633, −8.424956291227087854705975994943, −7.58493524026161590508768706991, −6.81777222133053344278430271644, −5.66602774536000681594820872108, −3.99888896709999327050707342580, −2.94289741066041825777760458380, −1.73288391344918494292290772918, 0,
1.73288391344918494292290772918, 2.94289741066041825777760458380, 3.99888896709999327050707342580, 5.66602774536000681594820872108, 6.81777222133053344278430271644, 7.58493524026161590508768706991, 8.424956291227087854705975994943, 8.842074896028910377868684174633, 9.562417468251677658992029589314