L(s) = 1 | − 2-s + 1.61·3-s − 4-s − 1.61·6-s + 1.23·7-s + 3·8-s − 0.381·9-s + 3.61·11-s − 1.61·12-s + 0.381·13-s − 1.23·14-s − 16-s + 3.85·17-s + 0.381·18-s + 0.618·19-s + 2.00·21-s − 3.61·22-s − 8.47·23-s + 4.85·24-s − 0.381·26-s − 5.47·27-s − 1.23·28-s − 7.09·29-s − 1.52·31-s − 5·32-s + 5.85·33-s − 3.85·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.934·3-s − 0.5·4-s − 0.660·6-s + 0.467·7-s + 1.06·8-s − 0.127·9-s + 1.09·11-s − 0.467·12-s + 0.105·13-s − 0.330·14-s − 0.250·16-s + 0.934·17-s + 0.0900·18-s + 0.141·19-s + 0.436·21-s − 0.771·22-s − 1.76·23-s + 0.990·24-s − 0.0749·26-s − 1.05·27-s − 0.233·28-s − 1.31·29-s − 0.274·31-s − 0.883·32-s + 1.01·33-s − 0.660·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 - 1.23T + 7T^{2} \) |
| 11 | \( 1 - 3.61T + 11T^{2} \) |
| 13 | \( 1 - 0.381T + 13T^{2} \) |
| 17 | \( 1 - 3.85T + 17T^{2} \) |
| 19 | \( 1 - 0.618T + 19T^{2} \) |
| 23 | \( 1 + 8.47T + 23T^{2} \) |
| 29 | \( 1 + 7.09T + 29T^{2} \) |
| 31 | \( 1 + 1.52T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 + 6.47T + 43T^{2} \) |
| 47 | \( 1 + 7.85T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 - 4.85T + 61T^{2} \) |
| 67 | \( 1 + 12.7T + 67T^{2} \) |
| 71 | \( 1 + 14.7T + 71T^{2} \) |
| 73 | \( 1 - 1.09T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
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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
−7.893288311115908960591392915004, −7.47035598278148838144068080098, −6.40295140482344952956253854917, −5.56067739061899013051516328548, −4.75054714234430280442058207794, −3.74254638290378031701523113178, −3.46493232716067488042819771296, −1.99330548440619093424174483893, −1.46516315221642415396202937985, 0,
1.46516315221642415396202937985, 1.99330548440619093424174483893, 3.46493232716067488042819771296, 3.74254638290378031701523113178, 4.75054714234430280442058207794, 5.56067739061899013051516328548, 6.40295140482344952956253854917, 7.47035598278148838144068080098, 7.893288311115908960591392915004