| L(s) = 1 | − 3-s + 9-s − 6·11-s + 7·13-s + 4·17-s − 5·19-s + 6·23-s − 27-s − 4·29-s − 8·31-s + 6·33-s + 37-s − 7·39-s − 2·41-s + 4·43-s − 8·47-s − 4·51-s + 8·53-s + 5·57-s − 5·61-s − 5·67-s − 6·69-s + 4·71-s − 73-s + 11·79-s + 81-s + 16·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s + 1.94·13-s + 0.970·17-s − 1.14·19-s + 1.25·23-s − 0.192·27-s − 0.742·29-s − 1.43·31-s + 1.04·33-s + 0.164·37-s − 1.12·39-s − 0.312·41-s + 0.609·43-s − 1.16·47-s − 0.560·51-s + 1.09·53-s + 0.662·57-s − 0.640·61-s − 0.610·67-s − 0.722·69-s + 0.474·71-s − 0.117·73-s + 1.23·79-s + 1/9·81-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 11 T + p T^{2} \) |
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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
−15.31839538921239, −15.11893020932706, −14.46388700032000, −13.56855432651346, −13.17921158608553, −12.94808060463604, −12.33591410166292, −11.57504532603599, −10.92054301033686, −10.72329762975760, −10.35971668973842, −9.429563453167634, −8.911010016828155, −8.269762310461257, −7.748912108806579, −7.228003259726823, −6.351787403974392, −5.995205000919868, −5.217679343086431, −5.016531747569951, −3.888473923841338, −3.501061023012147, −2.636962577515189, −1.789345095716412, −0.9594942724561404, 0,
0.9594942724561404, 1.789345095716412, 2.636962577515189, 3.501061023012147, 3.888473923841338, 5.016531747569951, 5.217679343086431, 5.995205000919868, 6.351787403974392, 7.228003259726823, 7.748912108806579, 8.269762310461257, 8.911010016828155, 9.429563453167634, 10.35971668973842, 10.72329762975760, 10.92054301033686, 11.57504532603599, 12.33591410166292, 12.94808060463604, 13.17921158608553, 13.56855432651346, 14.46388700032000, 15.11893020932706, 15.31839538921239
