L(s) = 1 | − 2-s − 1.12·3-s + 4-s − 3.70·5-s + 1.12·6-s − 7-s − 8-s − 1.73·9-s + 3.70·10-s + 4.57·11-s − 1.12·12-s − 5.55·13-s + 14-s + 4.17·15-s + 16-s + 1.43·17-s + 1.73·18-s − 3.55·19-s − 3.70·20-s + 1.12·21-s − 4.57·22-s + 5.55·23-s + 1.12·24-s + 8.72·25-s + 5.55·26-s + 5.32·27-s − 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.650·3-s + 0.5·4-s − 1.65·5-s + 0.459·6-s − 0.377·7-s − 0.353·8-s − 0.577·9-s + 1.17·10-s + 1.38·11-s − 0.325·12-s − 1.53·13-s + 0.267·14-s + 1.07·15-s + 0.250·16-s + 0.347·17-s + 0.408·18-s − 0.814·19-s − 0.828·20-s + 0.245·21-s − 0.976·22-s + 1.15·23-s + 0.229·24-s + 1.74·25-s + 1.08·26-s + 1.02·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4278295318\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4278295318\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 41 | \( 1 - T \) |
good | 3 | \( 1 + 1.12T + 3T^{2} \) |
| 5 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 + 5.55T + 13T^{2} \) |
| 17 | \( 1 - 1.43T + 17T^{2} \) |
| 19 | \( 1 + 3.55T + 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 - 6.40T + 29T^{2} \) |
| 31 | \( 1 + 0.172T + 31T^{2} \) |
| 37 | \( 1 - 0.252T + 37T^{2} \) |
| 43 | \( 1 + 6.42T + 43T^{2} \) |
| 47 | \( 1 - 8.11T + 47T^{2} \) |
| 53 | \( 1 + 8.41T + 53T^{2} \) |
| 59 | \( 1 - 9.87T + 59T^{2} \) |
| 61 | \( 1 - 14.3T + 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 - 1.38T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 12.1T + 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 9.58T + 97T^{2} \) |
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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
−10.88666376344184805062850950743, −9.870666663075032552776576265569, −8.856896110991864497084457984185, −8.160810837509841927419448417816, −7.06274955828220202435048597931, −6.60084233518333489883423935489, −5.12676978873747803931935904066, −4.04113978157380851187750642065, −2.85321761632502117640457185816, −0.63967045228762778715899693228,
0.63967045228762778715899693228, 2.85321761632502117640457185816, 4.04113978157380851187750642065, 5.12676978873747803931935904066, 6.60084233518333489883423935489, 7.06274955828220202435048597931, 8.160810837509841927419448417816, 8.856896110991864497084457984185, 9.870666663075032552776576265569, 10.88666376344184805062850950743