L(s) = 1 | − 2.13·2-s + 2.57·4-s + 0.343·5-s − 2.49·7-s − 1.22·8-s − 0.734·10-s − 0.158·11-s − 0.313·13-s + 5.32·14-s − 2.52·16-s + 4.67·17-s − 4.91·19-s + 0.883·20-s + 0.339·22-s − 23-s − 4.88·25-s + 0.670·26-s − 6.41·28-s + 29-s + 8.73·31-s + 7.85·32-s − 9.99·34-s − 0.855·35-s + 7.70·37-s + 10.5·38-s − 0.421·40-s − 7.50·41-s + ⋯ |
L(s) = 1 | − 1.51·2-s + 1.28·4-s + 0.153·5-s − 0.941·7-s − 0.433·8-s − 0.232·10-s − 0.0478·11-s − 0.0869·13-s + 1.42·14-s − 0.630·16-s + 1.13·17-s − 1.12·19-s + 0.197·20-s + 0.0723·22-s − 0.208·23-s − 0.976·25-s + 0.131·26-s − 1.21·28-s + 0.185·29-s + 1.56·31-s + 1.38·32-s − 1.71·34-s − 0.144·35-s + 1.26·37-s + 1.70·38-s − 0.0666·40-s − 1.17·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
good | 2 | \( 1 + 2.13T + 2T^{2} \) |
| 5 | \( 1 - 0.343T + 5T^{2} \) |
| 7 | \( 1 + 2.49T + 7T^{2} \) |
| 11 | \( 1 + 0.158T + 11T^{2} \) |
| 13 | \( 1 + 0.313T + 13T^{2} \) |
| 17 | \( 1 - 4.67T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 31 | \( 1 - 8.73T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + 7.50T + 41T^{2} \) |
| 43 | \( 1 - 4.03T + 43T^{2} \) |
| 47 | \( 1 + 3.87T + 47T^{2} \) |
| 53 | \( 1 + 5.89T + 53T^{2} \) |
| 59 | \( 1 - 7.19T + 59T^{2} \) |
| 61 | \( 1 - 10.4T + 61T^{2} \) |
| 67 | \( 1 + 5.79T + 67T^{2} \) |
| 71 | \( 1 - 0.837T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 6.13T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 + 3.81T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.025615194095920882573767740108, −7.18740296120410829117988429819, −6.43211944432209159739873901440, −6.00661242442028926395581208983, −4.85023700061230589721372240390, −3.91992499360950269874982141971, −2.92477426438490814097437580573, −2.10827275108834761505472869970, −1.04085261447633186565126080314, 0,
1.04085261447633186565126080314, 2.10827275108834761505472869970, 2.92477426438490814097437580573, 3.91992499360950269874982141971, 4.85023700061230589721372240390, 6.00661242442028926395581208983, 6.43211944432209159739873901440, 7.18740296120410829117988429819, 8.025615194095920882573767740108