| L(s) = 1 | + 2-s + 0.184·3-s + 4-s − 5-s + 0.184·6-s − 4.92·7-s + 8-s − 2.96·9-s − 10-s + 1.28·11-s + 0.184·12-s − 3.30·13-s − 4.92·14-s − 0.184·15-s + 16-s + 4.26·17-s − 2.96·18-s − 6.22·19-s − 20-s − 0.906·21-s + 1.28·22-s − 5.01·23-s + 0.184·24-s + 25-s − 3.30·26-s − 1.09·27-s − 4.92·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.106·3-s + 0.5·4-s − 0.447·5-s + 0.0751·6-s − 1.85·7-s + 0.353·8-s − 0.988·9-s − 0.316·10-s + 0.387·11-s + 0.0531·12-s − 0.917·13-s − 1.31·14-s − 0.0475·15-s + 0.250·16-s + 1.03·17-s − 0.699·18-s − 1.42·19-s − 0.223·20-s − 0.197·21-s + 0.274·22-s − 1.04·23-s + 0.0375·24-s + 0.200·25-s − 0.648·26-s − 0.211·27-s − 0.929·28-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
| \(L(1)\) |
\(\approx\) |
\(1.406075161\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.406075161\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.184T + 3T^{2} \) |
| 7 | \( 1 + 4.92T + 7T^{2} \) |
| 11 | \( 1 - 1.28T + 11T^{2} \) |
| 13 | \( 1 + 3.30T + 13T^{2} \) |
| 17 | \( 1 - 4.26T + 17T^{2} \) |
| 19 | \( 1 + 6.22T + 19T^{2} \) |
| 23 | \( 1 + 5.01T + 23T^{2} \) |
| 29 | \( 1 - 6.66T + 29T^{2} \) |
| 31 | \( 1 + 7.18T + 31T^{2} \) |
| 37 | \( 1 - 6.40T + 37T^{2} \) |
| 41 | \( 1 + 2.28T + 41T^{2} \) |
| 43 | \( 1 - 9.65T + 43T^{2} \) |
| 47 | \( 1 - 1.13T + 47T^{2} \) |
| 53 | \( 1 - 9.64T + 53T^{2} \) |
| 59 | \( 1 + 3.51T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 6.41T + 71T^{2} \) |
| 73 | \( 1 + 4.40T + 73T^{2} \) |
| 79 | \( 1 - 8.36T + 79T^{2} \) |
| 83 | \( 1 - 6.86T + 83T^{2} \) |
| 89 | \( 1 - 5.62T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−7.911088208241986343900473227437, −7.27891061566934186978269220174, −6.42109179580922911009783208898, −6.06718477811450115844193725832, −5.31521617631857794861672903788, −4.19108604081213987639733036807, −3.69273442220673672483719382336, −2.87697354731990097174355254309, −2.31915201354211322931353405410, −0.51675027524601747192351287944,
0.51675027524601747192351287944, 2.31915201354211322931353405410, 2.87697354731990097174355254309, 3.69273442220673672483719382336, 4.19108604081213987639733036807, 5.31521617631857794861672903788, 6.06718477811450115844193725832, 6.42109179580922911009783208898, 7.27891061566934186978269220174, 7.911088208241986343900473227437
