L(s) = 1 | − 1.18·2-s − 1.57·3-s − 0.588·4-s + 1.86·6-s − 7-s + 3.07·8-s − 0.533·9-s + 6.49·11-s + 0.924·12-s − 0.322·13-s + 1.18·14-s − 2.47·16-s − 5.08·17-s + 0.633·18-s − 0.234·19-s + 1.57·21-s − 7.71·22-s − 23-s − 4.82·24-s + 0.382·26-s + 5.54·27-s + 0.588·28-s − 4.41·29-s + 3.11·31-s − 3.20·32-s − 10.1·33-s + 6.04·34-s + ⋯ |
L(s) = 1 | − 0.840·2-s − 0.906·3-s − 0.294·4-s + 0.761·6-s − 0.377·7-s + 1.08·8-s − 0.177·9-s + 1.95·11-s + 0.266·12-s − 0.0893·13-s + 0.317·14-s − 0.618·16-s − 1.23·17-s + 0.149·18-s − 0.0538·19-s + 0.342·21-s − 1.64·22-s − 0.208·23-s − 0.985·24-s + 0.0750·26-s + 1.06·27-s + 0.111·28-s − 0.820·29-s + 0.559·31-s − 0.567·32-s − 1.77·33-s + 1.03·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 + 1.18T + 2T^{2} \) |
| 3 | \( 1 + 1.57T + 3T^{2} \) |
| 11 | \( 1 - 6.49T + 11T^{2} \) |
| 13 | \( 1 + 0.322T + 13T^{2} \) |
| 17 | \( 1 + 5.08T + 17T^{2} \) |
| 19 | \( 1 + 0.234T + 19T^{2} \) |
| 29 | \( 1 + 4.41T + 29T^{2} \) |
| 31 | \( 1 - 3.11T + 31T^{2} \) |
| 37 | \( 1 + 11.7T + 37T^{2} \) |
| 41 | \( 1 + 9.61T + 41T^{2} \) |
| 43 | \( 1 - 4.63T + 43T^{2} \) |
| 47 | \( 1 - 10.4T + 47T^{2} \) |
| 53 | \( 1 - 7.94T + 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 + 0.967T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 0.213T + 71T^{2} \) |
| 73 | \( 1 - 6.15T + 73T^{2} \) |
| 79 | \( 1 + 8.13T + 79T^{2} \) |
| 83 | \( 1 + 6.84T + 83T^{2} \) |
| 89 | \( 1 - 4.38T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.482536620861848485917464285413, −7.06822818961386553892819661939, −6.83349986380806907856875598935, −5.95996065858824304773445197290, −5.14897234357831207074953201384, −4.25309904657704497639112179786, −3.64287930509742575130533029741, −2.09811195003651209490650013818, −1.03579952761794639340325919959, 0,
1.03579952761794639340325919959, 2.09811195003651209490650013818, 3.64287930509742575130533029741, 4.25309904657704497639112179786, 5.14897234357831207074953201384, 5.95996065858824304773445197290, 6.83349986380806907856875598935, 7.06822818961386553892819661939, 8.482536620861848485917464285413