L(s) = 1 | − 2-s + 0.489·3-s + 4-s + 2.70·5-s − 0.489·6-s + 1.75·7-s − 8-s − 2.76·9-s − 2.70·10-s + 1.86·11-s + 0.489·12-s − 5.77·13-s − 1.75·14-s + 1.32·15-s + 16-s + 0.411·17-s + 2.76·18-s − 3.30·19-s + 2.70·20-s + 0.859·21-s − 1.86·22-s + 23-s − 0.489·24-s + 2.32·25-s + 5.77·26-s − 2.81·27-s + 1.75·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.282·3-s + 0.5·4-s + 1.21·5-s − 0.199·6-s + 0.664·7-s − 0.353·8-s − 0.920·9-s − 0.856·10-s + 0.561·11-s + 0.141·12-s − 1.60·13-s − 0.469·14-s + 0.342·15-s + 0.250·16-s + 0.0997·17-s + 0.650·18-s − 0.758·19-s + 0.605·20-s + 0.187·21-s − 0.397·22-s + 0.208·23-s − 0.0998·24-s + 0.465·25-s + 1.13·26-s − 0.542·27-s + 0.332·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
good | 3 | \( 1 - 0.489T + 3T^{2} \) |
| 5 | \( 1 - 2.70T + 5T^{2} \) |
| 7 | \( 1 - 1.75T + 7T^{2} \) |
| 11 | \( 1 - 1.86T + 11T^{2} \) |
| 13 | \( 1 + 5.77T + 13T^{2} \) |
| 17 | \( 1 - 0.411T + 17T^{2} \) |
| 19 | \( 1 + 3.30T + 19T^{2} \) |
| 29 | \( 1 + 8.95T + 29T^{2} \) |
| 31 | \( 1 - 4.04T + 31T^{2} \) |
| 37 | \( 1 - 8.37T + 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 + 4.95T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 - 8.33T + 53T^{2} \) |
| 59 | \( 1 - 13.2T + 59T^{2} \) |
| 61 | \( 1 - 9.35T + 61T^{2} \) |
| 67 | \( 1 - 6.28T + 67T^{2} \) |
| 71 | \( 1 + 4.46T + 71T^{2} \) |
| 73 | \( 1 + 13.5T + 73T^{2} \) |
| 79 | \( 1 + 7.13T + 79T^{2} \) |
| 83 | \( 1 - 6.71T + 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 + 9.59T + 97T^{2} \) |
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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
−7.956302115274510132569604908566, −7.00387104960306188711402868615, −6.46039971835462421016513288229, −5.52488734427702859521376534000, −5.13139162535518409750253785169, −3.99710160984130249887223166179, −2.80002033248492676262146920758, −2.21689463997384185224599920175, −1.50918940921191755528648737472, 0,
1.50918940921191755528648737472, 2.21689463997384185224599920175, 2.80002033248492676262146920758, 3.99710160984130249887223166179, 5.13139162535518409750253785169, 5.52488734427702859521376534000, 6.46039971835462421016513288229, 7.00387104960306188711402868615, 7.956302115274510132569604908566