| L(s) = 1 | + 2-s − 2.88·3-s + 4-s − 1.11·5-s − 2.88·6-s − 7-s + 8-s + 5.29·9-s − 1.11·10-s + 1.58·11-s − 2.88·12-s + 1.03·13-s − 14-s + 3.20·15-s + 16-s − 3.95·17-s + 5.29·18-s + 3.63·19-s − 1.11·20-s + 2.88·21-s + 1.58·22-s − 2.60·23-s − 2.88·24-s − 3.76·25-s + 1.03·26-s − 6.61·27-s − 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.66·3-s + 0.5·4-s − 0.497·5-s − 1.17·6-s − 0.377·7-s + 0.353·8-s + 1.76·9-s − 0.351·10-s + 0.477·11-s − 0.831·12-s + 0.287·13-s − 0.267·14-s + 0.827·15-s + 0.250·16-s − 0.959·17-s + 1.24·18-s + 0.834·19-s − 0.248·20-s + 0.628·21-s + 0.337·22-s − 0.544·23-s − 0.587·24-s − 0.752·25-s + 0.203·26-s − 1.27·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 431 | \( 1 - T \) |
| good | 3 | \( 1 + 2.88T + 3T^{2} \) |
| 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 - 1.58T + 11T^{2} \) |
| 13 | \( 1 - 1.03T + 13T^{2} \) |
| 17 | \( 1 + 3.95T + 17T^{2} \) |
| 19 | \( 1 - 3.63T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 - 0.755T + 29T^{2} \) |
| 31 | \( 1 + 7.11T + 31T^{2} \) |
| 37 | \( 1 - 5.29T + 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 - 6.15T + 47T^{2} \) |
| 53 | \( 1 - 3.71T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 7.20T + 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 3.27T + 71T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 - 14.0T + 79T^{2} \) |
| 83 | \( 1 - 2.97T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 + 11.5T + 97T^{2} \) |
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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
−7.36429830515755452188081927903, −6.77352572279059715610347151216, −6.19884931991464329696271978217, −5.58397159244580161454834857511, −4.95907312977150707662677380163, −4.07745578662681291834734327399, −3.67605202746341337523250545553, −2.30936128563608608535096585443, −1.13260192532082706645205190370, 0,
1.13260192532082706645205190370, 2.30936128563608608535096585443, 3.67605202746341337523250545553, 4.07745578662681291834734327399, 4.95907312977150707662677380163, 5.58397159244580161454834857511, 6.19884931991464329696271978217, 6.77352572279059715610347151216, 7.36429830515755452188081927903
