L(s) = 1 | + 1.15·3-s − 3.19·5-s − 7-s − 1.67·9-s + 11-s − 13-s − 3.67·15-s − 5.49·17-s − 5.11·19-s − 1.15·21-s + 1.48·23-s + 5.19·25-s − 5.38·27-s − 2.51·29-s − 7.49·31-s + 1.15·33-s + 3.19·35-s + 4.43·37-s − 1.15·39-s + 0.657·41-s + 6.68·43-s + 5.34·45-s + 2.89·47-s + 49-s − 6.32·51-s − 3.25·53-s − 3.19·55-s + ⋯ |
L(s) = 1 | + 0.664·3-s − 1.42·5-s − 0.377·7-s − 0.557·9-s + 0.301·11-s − 0.277·13-s − 0.949·15-s − 1.33·17-s − 1.17·19-s − 0.251·21-s + 0.310·23-s + 1.03·25-s − 1.03·27-s − 0.466·29-s − 1.34·31-s + 0.200·33-s + 0.539·35-s + 0.729·37-s − 0.184·39-s + 0.102·41-s + 1.01·43-s + 0.796·45-s + 0.422·47-s + 0.142·49-s − 0.885·51-s − 0.447·53-s − 0.430·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7176612532\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7176612532\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 1.15T + 3T^{2} \) |
| 5 | \( 1 + 3.19T + 5T^{2} \) |
| 17 | \( 1 + 5.49T + 17T^{2} \) |
| 19 | \( 1 + 5.11T + 19T^{2} \) |
| 23 | \( 1 - 1.48T + 23T^{2} \) |
| 29 | \( 1 + 2.51T + 29T^{2} \) |
| 31 | \( 1 + 7.49T + 31T^{2} \) |
| 37 | \( 1 - 4.43T + 37T^{2} \) |
| 41 | \( 1 - 0.657T + 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 - 2.89T + 47T^{2} \) |
| 53 | \( 1 + 3.25T + 53T^{2} \) |
| 59 | \( 1 - 3.56T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 7.93T + 67T^{2} \) |
| 71 | \( 1 - 6.97T + 71T^{2} \) |
| 73 | \( 1 + 5.60T + 73T^{2} \) |
| 79 | \( 1 - 8.60T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 9.04T + 89T^{2} \) |
| 97 | \( 1 - 7.78T + 97T^{2} \) |
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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
−7.77484957626831727397767527176, −7.37380489569205929080096173267, −6.55570637099665458424065975798, −5.87539377417642127391244226065, −4.77685258046455056922267983474, −4.10314145363929521144913855006, −3.60431259344981631976559639380, −2.75345059090556870366203915664, −1.99712889425619858512111131672, −0.37938414777856202988291825376,
0.37938414777856202988291825376, 1.99712889425619858512111131672, 2.75345059090556870366203915664, 3.60431259344981631976559639380, 4.10314145363929521144913855006, 4.77685258046455056922267983474, 5.87539377417642127391244226065, 6.55570637099665458424065975798, 7.37380489569205929080096173267, 7.77484957626831727397767527176