| L(s) = 1 | + 2-s − 1.67·3-s + 4-s − 0.572·5-s − 1.67·6-s − 0.229·7-s + 8-s − 0.196·9-s − 0.572·10-s − 11-s − 1.67·12-s + 1.39·13-s − 0.229·14-s + 0.958·15-s + 16-s − 2.96·17-s − 0.196·18-s + 6.23·19-s − 0.572·20-s + 0.383·21-s − 22-s + 0.283·23-s − 1.67·24-s − 4.67·25-s + 1.39·26-s + 5.35·27-s − 0.229·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.966·3-s + 0.5·4-s − 0.255·5-s − 0.683·6-s − 0.0866·7-s + 0.353·8-s − 0.0654·9-s − 0.180·10-s − 0.301·11-s − 0.483·12-s + 0.386·13-s − 0.0612·14-s + 0.247·15-s + 0.250·16-s − 0.717·17-s − 0.0462·18-s + 1.43·19-s − 0.127·20-s + 0.0837·21-s − 0.213·22-s + 0.0591·23-s − 0.341·24-s − 0.934·25-s + 0.273·26-s + 1.02·27-s − 0.0433·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 + 0.572T + 5T^{2} \) |
| 7 | \( 1 + 0.229T + 7T^{2} \) |
| 13 | \( 1 - 1.39T + 13T^{2} \) |
| 17 | \( 1 + 2.96T + 17T^{2} \) |
| 19 | \( 1 - 6.23T + 19T^{2} \) |
| 23 | \( 1 - 0.283T + 23T^{2} \) |
| 29 | \( 1 + 2.13T + 29T^{2} \) |
| 31 | \( 1 + 1.81T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 + 3.81T + 41T^{2} \) |
| 43 | \( 1 + 4.03T + 43T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 + 0.559T + 53T^{2} \) |
| 59 | \( 1 - 3.35T + 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 67 | \( 1 + 7.29T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 4.30T + 79T^{2} \) |
| 83 | \( 1 + 9.51T + 83T^{2} \) |
| 89 | \( 1 + 5.63T + 89T^{2} \) |
| 97 | \( 1 - 5.17T + 97T^{2} \) |
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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
−7.78553714917284003507162467914, −7.16365952194538971482383113784, −6.26306432537662421800074002100, −5.80689965492382557547607156640, −5.09084836327322654334000474306, −4.39192219592082940890500581460, −3.46678798309197635721494172696, −2.63980557859031600001510477181, −1.35240854075607246280378460776, 0,
1.35240854075607246280378460776, 2.63980557859031600001510477181, 3.46678798309197635721494172696, 4.39192219592082940890500581460, 5.09084836327322654334000474306, 5.80689965492382557547607156640, 6.26306432537662421800074002100, 7.16365952194538971482383113784, 7.78553714917284003507162467914
