| L(s) = 1 | + 2.41·3-s − 2.82·5-s + 4.41·7-s + 2.82·9-s + 3.24·11-s − 6.82·15-s − 5.82·17-s − 1.24·19-s + 10.6·21-s − 1.24·23-s + 3.00·25-s − 0.414·27-s + 8.65·29-s + 5.65·31-s + 7.82·33-s − 12.4·35-s + 7.48·37-s + 5.82·41-s − 4.07·43-s − 8·45-s + 6·47-s + 12.4·49-s − 14.0·51-s − 2.82·53-s − 9.17·55-s − 3·57-s − 1.24·59-s + ⋯ |
| L(s) = 1 | + 1.39·3-s − 1.26·5-s + 1.66·7-s + 0.942·9-s + 0.977·11-s − 1.76·15-s − 1.41·17-s − 0.285·19-s + 2.32·21-s − 0.259·23-s + 0.600·25-s − 0.0797·27-s + 1.60·29-s + 1.01·31-s + 1.36·33-s − 2.11·35-s + 1.23·37-s + 0.910·41-s − 0.620·43-s − 1.19·45-s + 0.875·47-s + 1.78·49-s − 1.97·51-s − 0.388·53-s − 1.23·55-s − 0.397·57-s − 0.161·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.385087613\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.385087613\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 - 3.24T + 11T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 + 1.24T + 19T^{2} \) |
| 23 | \( 1 + 1.24T + 23T^{2} \) |
| 29 | \( 1 - 8.65T + 29T^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 - 7.48T + 37T^{2} \) |
| 41 | \( 1 - 5.82T + 41T^{2} \) |
| 43 | \( 1 + 4.07T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 + 2.82T + 53T^{2} \) |
| 59 | \( 1 + 1.24T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 + 7.24T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 6T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 - 3.34T + 89T^{2} \) |
| 97 | \( 1 - 9T + 97T^{2} \) |
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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
−8.163002021581106903165300470532, −7.80260847911718824346767896848, −7.02670816124336143730025666275, −6.21158818162114220661713603136, −4.80414533850404788822905637947, −4.31453885098004132106142828463, −3.87484245944889210098693645298, −2.76205241891822037752605317653, −2.05182273091076588194428043791, −0.968780946651762960778269978958,
0.968780946651762960778269978958, 2.05182273091076588194428043791, 2.76205241891822037752605317653, 3.87484245944889210098693645298, 4.31453885098004132106142828463, 4.80414533850404788822905637947, 6.21158818162114220661713603136, 7.02670816124336143730025666275, 7.80260847911718824346767896848, 8.163002021581106903165300470532
