| L(s) = 1 | − 2-s + 3-s + 4-s + 0.266·5-s − 6-s + 1.01·7-s − 8-s + 9-s − 0.266·10-s + 3.25·11-s + 12-s − 1.28·13-s − 1.01·14-s + 0.266·15-s + 16-s − 1.67·17-s − 18-s + 19-s + 0.266·20-s + 1.01·21-s − 3.25·22-s + 5.10·23-s − 24-s − 4.92·25-s + 1.28·26-s + 27-s + 1.01·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 0.5·4-s + 0.119·5-s − 0.408·6-s + 0.382·7-s − 0.353·8-s + 0.333·9-s − 0.0842·10-s + 0.982·11-s + 0.288·12-s − 0.357·13-s − 0.270·14-s + 0.0687·15-s + 0.250·16-s − 0.406·17-s − 0.235·18-s + 0.229·19-s + 0.0595·20-s + 0.221·21-s − 0.694·22-s + 1.06·23-s − 0.204·24-s − 0.985·25-s + 0.252·26-s + 0.192·27-s + 0.191·28-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
| \(L(1)\) |
\(\approx\) |
\(2.120751491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.120751491\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.266T + 5T^{2} \) |
| 7 | \( 1 - 1.01T + 7T^{2} \) |
| 11 | \( 1 - 3.25T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 17 | \( 1 + 1.67T + 17T^{2} \) |
| 23 | \( 1 - 5.10T + 23T^{2} \) |
| 29 | \( 1 - 6.65T + 29T^{2} \) |
| 31 | \( 1 - 1.23T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 + 6.47T + 41T^{2} \) |
| 43 | \( 1 + 3.10T + 43T^{2} \) |
| 47 | \( 1 + 0.378T + 47T^{2} \) |
| 59 | \( 1 + 1.10T + 59T^{2} \) |
| 61 | \( 1 - 5.86T + 61T^{2} \) |
| 67 | \( 1 - 0.813T + 67T^{2} \) |
| 71 | \( 1 + 1.34T + 71T^{2} \) |
| 73 | \( 1 - 10.6T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 + 0.792T + 83T^{2} \) |
| 89 | \( 1 - 2.93T + 89T^{2} \) |
| 97 | \( 1 - 6.46T + 97T^{2} \) |
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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
−8.193497228844979076977829189818, −7.51819821486885764127360385925, −6.73812808889857003241755789028, −6.26615522201263135843956434809, −5.14023537085777962061490840155, −4.40539931580254486650363081935, −3.48792995597374293079748580082, −2.62854063198891538835015483850, −1.77524944121076511391778951686, −0.862650537124335924223102444700,
0.862650537124335924223102444700, 1.77524944121076511391778951686, 2.62854063198891538835015483850, 3.48792995597374293079748580082, 4.40539931580254486650363081935, 5.14023537085777962061490840155, 6.26615522201263135843956434809, 6.73812808889857003241755789028, 7.51819821486885764127360385925, 8.193497228844979076977829189818
