| L(s) = 1 | + 2-s − 1.27·3-s + 4-s − 0.808·5-s − 1.27·6-s + 7-s + 8-s − 1.36·9-s − 0.808·10-s − 1.04·11-s − 1.27·12-s + 4.47·13-s + 14-s + 1.03·15-s + 16-s − 2.56·17-s − 1.36·18-s + 0.0511·19-s − 0.808·20-s − 1.27·21-s − 1.04·22-s − 0.498·23-s − 1.27·24-s − 4.34·25-s + 4.47·26-s + 5.58·27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.738·3-s + 0.5·4-s − 0.361·5-s − 0.522·6-s + 0.377·7-s + 0.353·8-s − 0.454·9-s − 0.255·10-s − 0.316·11-s − 0.369·12-s + 1.24·13-s + 0.267·14-s + 0.266·15-s + 0.250·16-s − 0.621·17-s − 0.321·18-s + 0.0117·19-s − 0.180·20-s − 0.279·21-s − 0.223·22-s − 0.103·23-s − 0.261·24-s − 0.869·25-s + 0.877·26-s + 1.07·27-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 | \( 1 + 1.27T + 3T^{2} \) |
| 5 | \( 1 + 0.808T + 5T^{2} \) |
| 11 | \( 1 + 1.04T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 2.56T + 17T^{2} \) |
| 19 | \( 1 - 0.0511T + 19T^{2} \) |
| 23 | \( 1 + 0.498T + 23T^{2} \) |
| 29 | \( 1 - 0.297T + 29T^{2} \) |
| 31 | \( 1 + 1.84T + 31T^{2} \) |
| 37 | \( 1 + 3.42T + 37T^{2} \) |
| 41 | \( 1 + 7.76T + 41T^{2} \) |
| 43 | \( 1 - 7.95T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 - 7.04T + 59T^{2} \) |
| 61 | \( 1 - 4.41T + 61T^{2} \) |
| 67 | \( 1 + 8.94T + 67T^{2} \) |
| 71 | \( 1 + 2.64T + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 5.79T + 83T^{2} \) |
| 89 | \( 1 - 6.66T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.59594775723281518252532912273, −6.82755571658069860740424796880, −6.11636252275474587633268685070, −5.59069532351615483522635746141, −4.91510483600080278354200773898, −4.09645883450537704872936661314, −3.42794344522659504250298967984, −2.41877921664558148637481189782, −1.35233086132072267189903406929, 0,
1.35233086132072267189903406929, 2.41877921664558148637481189782, 3.42794344522659504250298967984, 4.09645883450537704872936661314, 4.91510483600080278354200773898, 5.59069532351615483522635746141, 6.11636252275474587633268685070, 6.82755571658069860740424796880, 7.59594775723281518252532912273
