L(s) = 1 | − 2-s + 3-s + 4-s − 1.39·5-s − 6-s − 2.74·7-s − 8-s + 9-s + 1.39·10-s + 2.62·11-s + 12-s − 5.97·13-s + 2.74·14-s − 1.39·15-s + 16-s + 2.56·17-s − 18-s + 19-s − 1.39·20-s − 2.74·21-s − 2.62·22-s + 3.23·23-s − 24-s − 3.06·25-s + 5.97·26-s + 27-s − 2.74·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.622·5-s − 0.408·6-s − 1.03·7-s − 0.353·8-s + 0.333·9-s + 0.440·10-s + 0.790·11-s + 0.288·12-s − 1.65·13-s + 0.732·14-s − 0.359·15-s + 0.250·16-s + 0.620·17-s − 0.235·18-s + 0.229·19-s − 0.311·20-s − 0.598·21-s − 0.558·22-s + 0.674·23-s − 0.204·24-s − 0.612·25-s + 1.17·26-s + 0.192·27-s − 0.518·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9665483860\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9665483860\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 53 | \( 1 - T \) |
good | 5 | \( 1 + 1.39T + 5T^{2} \) |
| 7 | \( 1 + 2.74T + 7T^{2} \) |
| 11 | \( 1 - 2.62T + 11T^{2} \) |
| 13 | \( 1 + 5.97T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 + 9.01T + 29T^{2} \) |
| 31 | \( 1 + 2.08T + 31T^{2} \) |
| 37 | \( 1 - 1.83T + 37T^{2} \) |
| 41 | \( 1 - 9.67T + 41T^{2} \) |
| 43 | \( 1 + 1.23T + 43T^{2} \) |
| 47 | \( 1 - 2.24T + 47T^{2} \) |
| 59 | \( 1 + 9.85T + 59T^{2} \) |
| 61 | \( 1 + 3.64T + 61T^{2} \) |
| 67 | \( 1 - 9.96T + 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 - 10.2T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 3.97T + 83T^{2} \) |
| 89 | \( 1 + 12.8T + 89T^{2} \) |
| 97 | \( 1 - 17.4T + 97T^{2} \) |
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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
−7.82799296618172594951373127333, −7.57139117154391979058162563938, −6.95431362942504997310173618868, −6.15575696970778556086345258276, −5.26368820146150157348180123943, −4.19307566323333026763170511381, −3.48863230550234053180170370608, −2.79001576029207118253937487178, −1.85537833127402459406334858517, −0.53968321294578239235854313435,
0.53968321294578239235854313435, 1.85537833127402459406334858517, 2.79001576029207118253937487178, 3.48863230550234053180170370608, 4.19307566323333026763170511381, 5.26368820146150157348180123943, 6.15575696970778556086345258276, 6.95431362942504997310173618868, 7.57139117154391979058162563938, 7.82799296618172594951373127333