L(s) = 1 | + 2.52·2-s + 3-s + 4.35·4-s − 3.63·5-s + 2.52·6-s − 7-s + 5.94·8-s + 9-s − 9.17·10-s + 5.28·11-s + 4.35·12-s − 6.11·13-s − 2.52·14-s − 3.63·15-s + 6.27·16-s + 0.479·17-s + 2.52·18-s − 7.37·19-s − 15.8·20-s − 21-s + 13.3·22-s − 5.10·23-s + 5.94·24-s + 8.23·25-s − 15.4·26-s + 27-s − 4.35·28-s + ⋯ |
L(s) = 1 | + 1.78·2-s + 0.577·3-s + 2.17·4-s − 1.62·5-s + 1.02·6-s − 0.377·7-s + 2.10·8-s + 0.333·9-s − 2.90·10-s + 1.59·11-s + 1.25·12-s − 1.69·13-s − 0.673·14-s − 0.939·15-s + 1.56·16-s + 0.116·17-s + 0.594·18-s − 1.69·19-s − 3.54·20-s − 0.218·21-s + 2.83·22-s − 1.06·23-s + 1.21·24-s + 1.64·25-s − 3.02·26-s + 0.192·27-s − 0.823·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8043 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 383 | \( 1 - T \) |
good | 2 | \( 1 - 2.52T + 2T^{2} \) |
| 5 | \( 1 + 3.63T + 5T^{2} \) |
| 11 | \( 1 - 5.28T + 11T^{2} \) |
| 13 | \( 1 + 6.11T + 13T^{2} \) |
| 17 | \( 1 - 0.479T + 17T^{2} \) |
| 19 | \( 1 + 7.37T + 19T^{2} \) |
| 23 | \( 1 + 5.10T + 23T^{2} \) |
| 29 | \( 1 - 10.6T + 29T^{2} \) |
| 31 | \( 1 + 8.71T + 31T^{2} \) |
| 37 | \( 1 + 7.60T + 37T^{2} \) |
| 41 | \( 1 - 9.57T + 41T^{2} \) |
| 43 | \( 1 + 8.45T + 43T^{2} \) |
| 47 | \( 1 + 1.79T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 8.44T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + 0.409T + 67T^{2} \) |
| 71 | \( 1 - 0.166T + 71T^{2} \) |
| 73 | \( 1 + 2.15T + 73T^{2} \) |
| 79 | \( 1 - 0.407T + 79T^{2} \) |
| 83 | \( 1 + 0.0289T + 83T^{2} \) |
| 89 | \( 1 - 6.57T + 89T^{2} \) |
| 97 | \( 1 + 2.70T + 97T^{2} \) |
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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
−7.22404706255672258787484311985, −6.71909706055195336708812593761, −6.21464266984789461055707581276, −4.99273897978499139510777548043, −4.42875026016005225929086213312, −3.97104792628279876023260944118, −3.43344775778500366065244841881, −2.65132744286106961847418119587, −1.76110933294303986358002384325, 0,
1.76110933294303986358002384325, 2.65132744286106961847418119587, 3.43344775778500366065244841881, 3.97104792628279876023260944118, 4.42875026016005225929086213312, 4.99273897978499139510777548043, 6.21464266984789461055707581276, 6.71909706055195336708812593761, 7.22404706255672258787484311985