| L(s) = 1 | − 2-s + 4-s − 3.54·5-s + 7-s − 8-s + 3.54·10-s + 0.458·13-s − 14-s + 16-s − 7.08·17-s + 4.54·19-s − 3.54·20-s + 0.541·23-s + 7.54·25-s − 0.458·26-s + 28-s − 3·29-s + 2.54·31-s − 32-s + 7.08·34-s − 3.54·35-s − 2.08·37-s − 4.54·38-s + 3.54·40-s + 9·41-s − 6.08·43-s − 0.541·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.58·5-s + 0.377·7-s − 0.353·8-s + 1.11·10-s + 0.127·13-s − 0.267·14-s + 0.250·16-s − 1.71·17-s + 1.04·19-s − 0.791·20-s + 0.112·23-s + 1.50·25-s − 0.0899·26-s + 0.188·28-s − 0.557·29-s + 0.456·31-s − 0.176·32-s + 1.21·34-s − 0.598·35-s − 0.342·37-s − 0.736·38-s + 0.559·40-s + 1.40·41-s − 0.927·43-s − 0.0798·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6534 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6629406865\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6629406865\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3.54T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 13 | \( 1 - 0.458T + 13T^{2} \) |
| 17 | \( 1 + 7.08T + 17T^{2} \) |
| 19 | \( 1 - 4.54T + 19T^{2} \) |
| 23 | \( 1 - 0.541T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 2.54T + 31T^{2} \) |
| 37 | \( 1 + 2.08T + 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 + 6.08T + 43T^{2} \) |
| 47 | \( 1 + 10.6T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 7.62T + 59T^{2} \) |
| 61 | \( 1 + 6.08T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 + 8.54T + 73T^{2} \) |
| 79 | \( 1 - 14.6T + 79T^{2} \) |
| 83 | \( 1 - 5.45T + 83T^{2} \) |
| 89 | \( 1 - 6.54T + 89T^{2} \) |
| 97 | \( 1 + 18.1T + 97T^{2} \) |
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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
−8.051257758963165027511904587440, −7.39857094583127902457466990161, −6.93126376030489238818523051405, −6.08701299431808619607653851783, −4.98875646494218151715444955228, −4.36669556520208899444270864451, −3.57211608651492907148695368927, −2.76531342739759110521177575226, −1.63751577180054240824252914138, −0.47044060991732011350416951087,
0.47044060991732011350416951087, 1.63751577180054240824252914138, 2.76531342739759110521177575226, 3.57211608651492907148695368927, 4.36669556520208899444270864451, 4.98875646494218151715444955228, 6.08701299431808619607653851783, 6.93126376030489238818523051405, 7.39857094583127902457466990161, 8.051257758963165027511904587440
