| L(s) = 1 | − 2.40·2-s + 3.79·4-s + 3.50·5-s + 5.09·7-s − 4.32·8-s − 8.42·10-s − 5.50·11-s + 3.91·13-s − 12.2·14-s + 2.81·16-s + 1.79·17-s + 3.82·19-s + 13.2·20-s + 13.2·22-s + 23-s + 7.25·25-s − 9.42·26-s + 19.3·28-s − 29-s − 6.41·31-s + 1.86·32-s − 4.31·34-s + 17.8·35-s − 6.13·37-s − 9.20·38-s − 15.1·40-s + 5.93·41-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.89·4-s + 1.56·5-s + 1.92·7-s − 1.52·8-s − 2.66·10-s − 1.65·11-s + 1.08·13-s − 3.28·14-s + 0.703·16-s + 0.434·17-s + 0.877·19-s + 2.97·20-s + 2.82·22-s + 0.208·23-s + 1.45·25-s − 1.84·26-s + 3.65·28-s − 0.185·29-s − 1.15·31-s + 0.330·32-s − 0.739·34-s + 3.01·35-s − 1.00·37-s − 1.49·38-s − 2.39·40-s + 0.927·41-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
| \(L(1)\) |
\(\approx\) |
\(1.603797769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.603797769\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.40T + 2T^{2} \) |
| 5 | \( 1 - 3.50T + 5T^{2} \) |
| 7 | \( 1 - 5.09T + 7T^{2} \) |
| 11 | \( 1 + 5.50T + 11T^{2} \) |
| 13 | \( 1 - 3.91T + 13T^{2} \) |
| 17 | \( 1 - 1.79T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 31 | \( 1 + 6.41T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 - 5.93T + 41T^{2} \) |
| 43 | \( 1 + 3.52T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 1.81T + 53T^{2} \) |
| 59 | \( 1 + 5.37T + 59T^{2} \) |
| 61 | \( 1 - 5.57T + 61T^{2} \) |
| 67 | \( 1 + 7.94T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 - 2.86T + 89T^{2} \) |
| 97 | \( 1 + 6.80T + 97T^{2} \) |
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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
−8.237192978702176402338795180665, −7.64898636022522334738762035104, −7.01809786717146839633201043743, −5.93150207789197390844077079897, −5.39478308471987542747857919078, −4.83283630720094150799826730486, −3.19773833728778925816814612959, −2.09415830038284417322337960878, −1.77269510810078445831037800768, −0.915673124340755784387419539353,
0.915673124340755784387419539353, 1.77269510810078445831037800768, 2.09415830038284417322337960878, 3.19773833728778925816814612959, 4.83283630720094150799826730486, 5.39478308471987542747857919078, 5.93150207789197390844077079897, 7.01809786717146839633201043743, 7.64898636022522334738762035104, 8.237192978702176402338795180665
