L(s) = 1 | + 0.823·3-s − 1.53·5-s + 0.257·7-s − 2.32·9-s + 1.07·11-s − 6.44·13-s − 1.26·15-s + 3.61·17-s − 2.46·19-s + 0.212·21-s + 6.43·23-s − 2.64·25-s − 4.38·27-s + 6.41·29-s − 4.45·31-s + 0.884·33-s − 0.395·35-s − 5.37·37-s − 5.30·39-s − 1.12·41-s + 5.77·43-s + 3.56·45-s + 10.2·47-s − 6.93·49-s + 2.97·51-s − 3.86·53-s − 1.64·55-s + ⋯ |
L(s) = 1 | + 0.475·3-s − 0.685·5-s + 0.0975·7-s − 0.773·9-s + 0.323·11-s − 1.78·13-s − 0.326·15-s + 0.875·17-s − 0.565·19-s + 0.0463·21-s + 1.34·23-s − 0.529·25-s − 0.843·27-s + 1.19·29-s − 0.799·31-s + 0.153·33-s − 0.0668·35-s − 0.883·37-s − 0.849·39-s − 0.176·41-s + 0.880·43-s + 0.530·45-s + 1.49·47-s − 0.990·49-s + 0.416·51-s − 0.530·53-s − 0.222·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.435419175\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.435419175\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 + T \) |
good | 3 | \( 1 - 0.823T + 3T^{2} \) |
| 5 | \( 1 + 1.53T + 5T^{2} \) |
| 7 | \( 1 - 0.257T + 7T^{2} \) |
| 11 | \( 1 - 1.07T + 11T^{2} \) |
| 13 | \( 1 + 6.44T + 13T^{2} \) |
| 17 | \( 1 - 3.61T + 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 - 6.43T + 23T^{2} \) |
| 29 | \( 1 - 6.41T + 29T^{2} \) |
| 31 | \( 1 + 4.45T + 31T^{2} \) |
| 37 | \( 1 + 5.37T + 37T^{2} \) |
| 41 | \( 1 + 1.12T + 41T^{2} \) |
| 43 | \( 1 - 5.77T + 43T^{2} \) |
| 47 | \( 1 - 10.2T + 47T^{2} \) |
| 53 | \( 1 + 3.86T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 5.43T + 61T^{2} \) |
| 67 | \( 1 + 3.75T + 67T^{2} \) |
| 71 | \( 1 - 1.92T + 71T^{2} \) |
| 73 | \( 1 + 9.39T + 73T^{2} \) |
| 79 | \( 1 + 3.75T + 79T^{2} \) |
| 83 | \( 1 - 7.37T + 83T^{2} \) |
| 89 | \( 1 + 2.01T + 89T^{2} \) |
| 97 | \( 1 - 0.889T + 97T^{2} \) |
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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
−8.070481609578446577801398323350, −7.39285684159969416339188322429, −6.95186413099779849848811336062, −5.85564020576909768474244835140, −5.14666050320657701735822605079, −4.43606505833640439078488434395, −3.52039559049303967985960381396, −2.84983571730022023561229195006, −2.04309350738226895027188660685, −0.59007097301929066974401161657,
0.59007097301929066974401161657, 2.04309350738226895027188660685, 2.84983571730022023561229195006, 3.52039559049303967985960381396, 4.43606505833640439078488434395, 5.14666050320657701735822605079, 5.85564020576909768474244835140, 6.95186413099779849848811336062, 7.39285684159969416339188322429, 8.070481609578446577801398323350