| L(s) = 1 | − 7-s + 3·11-s − 4·13-s − 2·19-s − 3·23-s − 9·29-s − 8·31-s + 5·37-s + 6·41-s − 11·43-s + 6·47-s + 49-s − 6·53-s − 10·61-s − 5·67-s + 15·71-s − 10·73-s − 3·77-s + 7·79-s + 12·83-s + 12·89-s + 4·91-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.904·11-s − 1.10·13-s − 0.458·19-s − 0.625·23-s − 1.67·29-s − 1.43·31-s + 0.821·37-s + 0.937·41-s − 1.67·43-s + 0.875·47-s + 1/7·49-s − 0.824·53-s − 1.28·61-s − 0.610·67-s + 1.78·71-s − 1.17·73-s − 0.341·77-s + 0.787·79-s + 1.31·83-s + 1.27·89-s + 0.419·91-s + 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.174264735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.174264735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 11 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 15 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−15.13723702474672, −14.75395997551984, −14.54322241157070, −13.68420240471687, −13.23270841508774, −12.60092443063324, −12.17138682873158, −11.66776650030746, −10.95132387071477, −10.56614130713623, −9.665102889230700, −9.386494806360080, −8.981800685040254, −7.995971090212130, −7.606124159765658, −6.946683108662961, −6.379124362353447, −5.758155187941822, −5.128283938664786, −4.329978922997544, −3.806600563376623, −3.119421667444041, −2.189249903288266, −1.656112682910114, −0.4133028610189962,
0.4133028610189962, 1.656112682910114, 2.189249903288266, 3.119421667444041, 3.806600563376623, 4.329978922997544, 5.128283938664786, 5.758155187941822, 6.379124362353447, 6.946683108662961, 7.606124159765658, 7.995971090212130, 8.981800685040254, 9.386494806360080, 9.665102889230700, 10.56614130713623, 10.95132387071477, 11.66776650030746, 12.17138682873158, 12.60092443063324, 13.23270841508774, 13.68420240471687, 14.54322241157070, 14.75395997551984, 15.13723702474672
