| L(s) = 1 | + 3-s − 1.84·5-s − 2.79·7-s + 9-s + 1.31·11-s + 2.49·13-s − 1.84·15-s − 3.40·17-s + 2.66·19-s − 2.79·21-s + 23-s − 1.60·25-s + 27-s + 29-s − 7.42·31-s + 1.31·33-s + 5.13·35-s + 2.47·37-s + 2.49·39-s + 5.01·41-s + 2.20·43-s − 1.84·45-s + 10.0·47-s + 0.789·49-s − 3.40·51-s − 0.225·53-s − 2.41·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.823·5-s − 1.05·7-s + 0.333·9-s + 0.395·11-s + 0.692·13-s − 0.475·15-s − 0.825·17-s + 0.612·19-s − 0.609·21-s + 0.208·23-s − 0.321·25-s + 0.192·27-s + 0.185·29-s − 1.33·31-s + 0.228·33-s + 0.868·35-s + 0.406·37-s + 0.399·39-s + 0.782·41-s + 0.335·43-s − 0.274·45-s + 1.47·47-s + 0.112·49-s − 0.476·51-s − 0.0309·53-s − 0.325·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 5 | \( 1 + 1.84T + 5T^{2} \) |
| 7 | \( 1 + 2.79T + 7T^{2} \) |
| 11 | \( 1 - 1.31T + 11T^{2} \) |
| 13 | \( 1 - 2.49T + 13T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 - 2.66T + 19T^{2} \) |
| 31 | \( 1 + 7.42T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 - 5.01T + 41T^{2} \) |
| 43 | \( 1 - 2.20T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 + 0.225T + 53T^{2} \) |
| 59 | \( 1 + 5.31T + 59T^{2} \) |
| 61 | \( 1 - 11.3T + 61T^{2} \) |
| 67 | \( 1 - 4.47T + 67T^{2} \) |
| 71 | \( 1 + 3.68T + 71T^{2} \) |
| 73 | \( 1 + 9.11T + 73T^{2} \) |
| 79 | \( 1 + 1.28T + 79T^{2} \) |
| 83 | \( 1 + 13.6T + 83T^{2} \) |
| 89 | \( 1 + 16.0T + 89T^{2} \) |
| 97 | \( 1 + 0.382T + 97T^{2} \) |
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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
−7.39310520589388870404730670226, −6.99573616733824058535401267768, −6.17594363359484094685207830444, −5.50487267139843380674341667125, −4.28760123666462986978779131053, −3.90198079681038360575980452291, −3.19886378678959776244596207190, −2.42014834836775487118337779404, −1.21180081081801366025377704891, 0,
1.21180081081801366025377704891, 2.42014834836775487118337779404, 3.19886378678959776244596207190, 3.90198079681038360575980452291, 4.28760123666462986978779131053, 5.50487267139843380674341667125, 6.17594363359484094685207830444, 6.99573616733824058535401267768, 7.39310520589388870404730670226
