| L(s) = 1 | + 2-s + 2.52·3-s + 4-s + 1.42·5-s + 2.52·6-s + 7-s + 8-s + 3.36·9-s + 1.42·10-s + 1.22·11-s + 2.52·12-s + 2.87·13-s + 14-s + 3.58·15-s + 16-s + 3.92·17-s + 3.36·18-s + 1.42·20-s + 2.52·21-s + 1.22·22-s − 3.82·23-s + 2.52·24-s − 2.98·25-s + 2.87·26-s + 0.913·27-s + 28-s − 1.85·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.45·3-s + 0.5·4-s + 0.635·5-s + 1.02·6-s + 0.377·7-s + 0.353·8-s + 1.12·9-s + 0.449·10-s + 0.368·11-s + 0.728·12-s + 0.798·13-s + 0.267·14-s + 0.925·15-s + 0.250·16-s + 0.951·17-s + 0.792·18-s + 0.317·20-s + 0.550·21-s + 0.260·22-s − 0.796·23-s + 0.514·24-s − 0.596·25-s + 0.564·26-s + 0.175·27-s + 0.188·28-s − 0.344·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.774905045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.774905045\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2.52T + 3T^{2} \) |
| 5 | \( 1 - 1.42T + 5T^{2} \) |
| 11 | \( 1 - 1.22T + 11T^{2} \) |
| 13 | \( 1 - 2.87T + 13T^{2} \) |
| 17 | \( 1 - 3.92T + 17T^{2} \) |
| 23 | \( 1 + 3.82T + 23T^{2} \) |
| 29 | \( 1 + 1.85T + 29T^{2} \) |
| 31 | \( 1 + 4.99T + 31T^{2} \) |
| 37 | \( 1 + 0.841T + 37T^{2} \) |
| 41 | \( 1 - 7.37T + 41T^{2} \) |
| 43 | \( 1 + 4.80T + 43T^{2} \) |
| 47 | \( 1 - 6.55T + 47T^{2} \) |
| 53 | \( 1 - 3.68T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 13.9T + 61T^{2} \) |
| 67 | \( 1 + 9.35T + 67T^{2} \) |
| 71 | \( 1 - 9.04T + 71T^{2} \) |
| 73 | \( 1 + 4.26T + 73T^{2} \) |
| 79 | \( 1 + 1.87T + 79T^{2} \) |
| 83 | \( 1 + 3.84T + 83T^{2} \) |
| 89 | \( 1 - 18.4T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−8.072199684119871973653441795365, −7.71041384118449288546972758637, −6.83006664992395322140180015424, −5.88187895789188215410345673898, −5.44868008017565990980151219728, −4.17736834438346434348123148068, −3.75710988852325522403659470038, −2.91799296932183364607204287231, −2.06956016977621900552410398895, −1.40074981971049265504399691178,
1.40074981971049265504399691178, 2.06956016977621900552410398895, 2.91799296932183364607204287231, 3.75710988852325522403659470038, 4.17736834438346434348123148068, 5.44868008017565990980151219728, 5.88187895789188215410345673898, 6.83006664992395322140180015424, 7.71041384118449288546972758637, 8.072199684119871973653441795365
