L(s) = 1 | + 2-s + 3-s + 4-s − 3.96·5-s + 6-s + 0.133·7-s + 8-s + 9-s − 3.96·10-s + 0.829·11-s + 12-s − 13-s + 0.133·14-s − 3.96·15-s + 16-s − 7.00·17-s + 18-s + 2.69·19-s − 3.96·20-s + 0.133·21-s + 0.829·22-s + 2.07·23-s + 24-s + 10.7·25-s − 26-s + 27-s + 0.133·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.77·5-s + 0.408·6-s + 0.0503·7-s + 0.353·8-s + 0.333·9-s − 1.25·10-s + 0.250·11-s + 0.288·12-s − 0.277·13-s + 0.0355·14-s − 1.02·15-s + 0.250·16-s − 1.69·17-s + 0.235·18-s + 0.617·19-s − 0.886·20-s + 0.0290·21-s + 0.176·22-s + 0.431·23-s + 0.204·24-s + 2.14·25-s − 0.196·26-s + 0.192·27-s + 0.0251·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 103 | \( 1 + T \) |
good | 5 | \( 1 + 3.96T + 5T^{2} \) |
| 7 | \( 1 - 0.133T + 7T^{2} \) |
| 11 | \( 1 - 0.829T + 11T^{2} \) |
| 17 | \( 1 + 7.00T + 17T^{2} \) |
| 19 | \( 1 - 2.69T + 19T^{2} \) |
| 23 | \( 1 - 2.07T + 23T^{2} \) |
| 29 | \( 1 - 0.727T + 29T^{2} \) |
| 31 | \( 1 - 7.57T + 31T^{2} \) |
| 37 | \( 1 + 6.98T + 37T^{2} \) |
| 41 | \( 1 + 0.446T + 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 - 5.80T + 47T^{2} \) |
| 53 | \( 1 + 8.83T + 53T^{2} \) |
| 59 | \( 1 + 10.1T + 59T^{2} \) |
| 61 | \( 1 - 5.99T + 61T^{2} \) |
| 67 | \( 1 + 13.4T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 8.51T + 73T^{2} \) |
| 79 | \( 1 + 7.01T + 79T^{2} \) |
| 83 | \( 1 + 1.00T + 83T^{2} \) |
| 89 | \( 1 - 4.77T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.52174866854197251490571555760, −6.88707873940477921118287428772, −6.27131956739279936600693906216, −5.04587084701662165700045421850, −4.42609781864400117556146671170, −4.06035021404294726145354278551, −3.14222599464247575000522688956, −2.64576888824869862683022427214, −1.34816046001889723365295458486, 0,
1.34816046001889723365295458486, 2.64576888824869862683022427214, 3.14222599464247575000522688956, 4.06035021404294726145354278551, 4.42609781864400117556146671170, 5.04587084701662165700045421850, 6.27131956739279936600693906216, 6.88707873940477921118287428772, 7.52174866854197251490571555760