L(s) = 1 | + 2-s − 3.30·3-s + 4-s − 0.0759·5-s − 3.30·6-s − 4.78·7-s + 8-s + 7.92·9-s − 0.0759·10-s + 11-s − 3.30·12-s − 2.77·13-s − 4.78·14-s + 0.251·15-s + 16-s − 0.186·17-s + 7.92·18-s − 0.0759·20-s + 15.8·21-s + 22-s − 4.77·23-s − 3.30·24-s − 4.99·25-s − 2.77·26-s − 16.2·27-s − 4.78·28-s + 10.7·29-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.90·3-s + 0.5·4-s − 0.0339·5-s − 1.34·6-s − 1.80·7-s + 0.353·8-s + 2.64·9-s − 0.0240·10-s + 0.301·11-s − 0.954·12-s − 0.770·13-s − 1.27·14-s + 0.0648·15-s + 0.250·16-s − 0.0452·17-s + 1.86·18-s − 0.0169·20-s + 3.45·21-s + 0.213·22-s − 0.995·23-s − 0.674·24-s − 0.998·25-s − 0.544·26-s − 3.13·27-s − 0.904·28-s + 1.99·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7942 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 3.30T + 3T^{2} \) |
| 5 | \( 1 + 0.0759T + 5T^{2} \) |
| 7 | \( 1 + 4.78T + 7T^{2} \) |
| 13 | \( 1 + 2.77T + 13T^{2} \) |
| 17 | \( 1 + 0.186T + 17T^{2} \) |
| 23 | \( 1 + 4.77T + 23T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 - 4.05T + 31T^{2} \) |
| 37 | \( 1 + 1.01T + 37T^{2} \) |
| 41 | \( 1 - 9.90T + 41T^{2} \) |
| 43 | \( 1 - 6.50T + 43T^{2} \) |
| 47 | \( 1 + 2.51T + 47T^{2} \) |
| 53 | \( 1 - 4.25T + 53T^{2} \) |
| 59 | \( 1 - 0.727T + 59T^{2} \) |
| 61 | \( 1 + 11.4T + 61T^{2} \) |
| 67 | \( 1 + 8.33T + 67T^{2} \) |
| 71 | \( 1 + 6.61T + 71T^{2} \) |
| 73 | \( 1 + 1.97T + 73T^{2} \) |
| 79 | \( 1 - 7.64T + 79T^{2} \) |
| 83 | \( 1 + 3.73T + 83T^{2} \) |
| 89 | \( 1 + 7.68T + 89T^{2} \) |
| 97 | \( 1 - 3.47T + 97T^{2} \) |
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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
−7.12275341824116325969238916385, −6.47247909081030896547276796820, −6.04927455358057894590186398009, −5.69371803885386056785120214653, −4.56481510961077395356540282753, −4.28717681025604589402718189270, −3.28830281304737859597550950031, −2.31718505712939734505951837204, −0.945135623970580511951953551366, 0,
0.945135623970580511951953551366, 2.31718505712939734505951837204, 3.28830281304737859597550950031, 4.28717681025604589402718189270, 4.56481510961077395356540282753, 5.69371803885386056785120214653, 6.04927455358057894590186398009, 6.47247909081030896547276796820, 7.12275341824116325969238916385