| L(s) = 1 | + 2.39·2-s − 3·3-s − 2.26·4-s + 5.91·5-s − 7.18·6-s − 28.4·7-s − 24.5·8-s + 9·9-s + 14.1·10-s + 7.44·11-s + 6.79·12-s + 41.9·13-s − 68.0·14-s − 17.7·15-s − 40.7·16-s + 95.9·17-s + 21.5·18-s − 163.·19-s − 13.3·20-s + 85.2·21-s + 17.8·22-s − 76.9·23-s + 73.7·24-s − 90.0·25-s + 100.·26-s − 27·27-s + 64.3·28-s + ⋯ |
| L(s) = 1 | + 0.846·2-s − 0.577·3-s − 0.282·4-s + 0.528·5-s − 0.488·6-s − 1.53·7-s − 1.08·8-s + 0.333·9-s + 0.447·10-s + 0.204·11-s + 0.163·12-s + 0.895·13-s − 1.29·14-s − 0.305·15-s − 0.637·16-s + 1.36·17-s + 0.282·18-s − 1.96·19-s − 0.149·20-s + 0.886·21-s + 0.172·22-s − 0.697·23-s + 0.627·24-s − 0.720·25-s + 0.758·26-s − 0.192·27-s + 0.434·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1011 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.462900378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.462900378\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 337 | \( 1 + 337T \) |
| good | 2 | \( 1 - 2.39T + 8T^{2} \) |
| 5 | \( 1 - 5.91T + 125T^{2} \) |
| 7 | \( 1 + 28.4T + 343T^{2} \) |
| 11 | \( 1 - 7.44T + 1.33e3T^{2} \) |
| 13 | \( 1 - 41.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 95.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 163.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 76.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 286.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 98.9T + 2.97e4T^{2} \) |
| 37 | \( 1 - 319.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 155.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 493.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 93.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 400.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 783.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 685.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 649.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 334.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 718.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 300.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 246.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.11e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 789.T + 9.12e5T^{2} \) |
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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
−9.569665727135963462776568602890, −9.062856168910238282895537337884, −7.81357884238850343823684172897, −6.50056556119485413749650379236, −5.98924307789219156887730937970, −5.55125843198945334603880978304, −4.02242705463765522651824906798, −3.69589269970312923335663841308, −2.29983370518031944479491258228, −0.56450870120334073902932372197,
0.56450870120334073902932372197, 2.29983370518031944479491258228, 3.69589269970312923335663841308, 4.02242705463765522651824906798, 5.55125843198945334603880978304, 5.98924307789219156887730937970, 6.50056556119485413749650379236, 7.81357884238850343823684172897, 9.062856168910238282895537337884, 9.569665727135963462776568602890
