L(s) = 1 | − 2-s − 3.13·3-s + 4-s − 5-s + 3.13·6-s + 1.27·7-s − 8-s + 6.80·9-s + 10-s + 3.13·11-s − 3.13·12-s − 7.04·13-s − 1.27·14-s + 3.13·15-s + 16-s + 2.66·17-s − 6.80·18-s + 1.42·19-s − 20-s − 4.00·21-s − 3.13·22-s + 7.11·23-s + 3.13·24-s + 25-s + 7.04·26-s − 11.9·27-s + 1.27·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.80·3-s + 0.5·4-s − 0.447·5-s + 1.27·6-s + 0.483·7-s − 0.353·8-s + 2.26·9-s + 0.316·10-s + 0.946·11-s − 0.903·12-s − 1.95·13-s − 0.341·14-s + 0.808·15-s + 0.250·16-s + 0.645·17-s − 1.60·18-s + 0.327·19-s − 0.223·20-s − 0.873·21-s − 0.669·22-s + 1.48·23-s + 0.639·24-s + 0.200·25-s + 1.38·26-s − 2.29·27-s + 0.241·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
good | 3 | \( 1 + 3.13T + 3T^{2} \) |
| 7 | \( 1 - 1.27T + 7T^{2} \) |
| 11 | \( 1 - 3.13T + 11T^{2} \) |
| 13 | \( 1 + 7.04T + 13T^{2} \) |
| 17 | \( 1 - 2.66T + 17T^{2} \) |
| 19 | \( 1 - 1.42T + 19T^{2} \) |
| 23 | \( 1 - 7.11T + 23T^{2} \) |
| 29 | \( 1 + 7.02T + 29T^{2} \) |
| 31 | \( 1 - 5.10T + 31T^{2} \) |
| 37 | \( 1 + 7.90T + 37T^{2} \) |
| 41 | \( 1 + 1.24T + 41T^{2} \) |
| 43 | \( 1 + 6.29T + 43T^{2} \) |
| 47 | \( 1 + 8.64T + 47T^{2} \) |
| 53 | \( 1 - 11.5T + 53T^{2} \) |
| 59 | \( 1 + 5.50T + 59T^{2} \) |
| 61 | \( 1 - 0.516T + 61T^{2} \) |
| 67 | \( 1 - 10.6T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 0.345T + 73T^{2} \) |
| 79 | \( 1 - 0.547T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 5.49T + 89T^{2} \) |
| 97 | \( 1 - 10.3T + 97T^{2} \) |
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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
−7.51405437757058649063691657209, −6.91724118787851801809077974260, −6.61712637458041943347555753165, −5.32643822825367950189606138981, −5.18785820987377848781612178995, −4.32103400659187335666782920132, −3.24602952812418463035609911835, −1.86358583014607814974024057438, −0.980853635854735403720433728875, 0,
0.980853635854735403720433728875, 1.86358583014607814974024057438, 3.24602952812418463035609911835, 4.32103400659187335666782920132, 5.18785820987377848781612178995, 5.32643822825367950189606138981, 6.61712637458041943347555753165, 6.91724118787851801809077974260, 7.51405437757058649063691657209