| L(s) = 1 | + 2·2-s − 3.19·3-s + 4·4-s − 6.38·6-s + 2.29·7-s + 8·8-s − 16.8·9-s + 7.85·11-s − 12.7·12-s − 45.9·13-s + 4.59·14-s + 16·16-s − 24.9·17-s − 33.6·18-s − 66.5·19-s − 7.34·21-s + 15.7·22-s + 144.·23-s − 25.5·24-s − 91.8·26-s + 139.·27-s + 9.19·28-s + 50.6·29-s + 151.·31-s + 32·32-s − 25.0·33-s − 49.9·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.614·3-s + 0.5·4-s − 0.434·6-s + 0.124·7-s + 0.353·8-s − 0.622·9-s + 0.215·11-s − 0.307·12-s − 0.980·13-s + 0.0877·14-s + 0.250·16-s − 0.356·17-s − 0.440·18-s − 0.803·19-s − 0.0763·21-s + 0.152·22-s + 1.31·23-s − 0.217·24-s − 0.693·26-s + 0.997·27-s + 0.0620·28-s + 0.324·29-s + 0.875·31-s + 0.176·32-s − 0.132·33-s − 0.251·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1250 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.223898563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.223898563\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 3.19T + 27T^{2} \) |
| 7 | \( 1 - 2.29T + 343T^{2} \) |
| 11 | \( 1 - 7.85T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 24.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 66.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 144.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 50.6T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 376.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 481.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 242.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 141.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 125.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 342.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 704.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 334.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.36e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 545.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 181.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 472.T + 9.12e5T^{2} \) |
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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
−9.311980595654633012414428440596, −8.516126097010625573302750291089, −7.44872971608824885714737401695, −6.65918748105796637471676482468, −5.91376915729786402752517484978, −5.02316358529743729891645287071, −4.41598170392580345326081921510, −3.12051800148429874648380138090, −2.22022243282390868541617821554, −0.68472086528532450155186025340,
0.68472086528532450155186025340, 2.22022243282390868541617821554, 3.12051800148429874648380138090, 4.41598170392580345326081921510, 5.02316358529743729891645287071, 5.91376915729786402752517484978, 6.65918748105796637471676482468, 7.44872971608824885714737401695, 8.516126097010625573302750291089, 9.311980595654633012414428440596
