| L(s) = 1 | − 5-s − 7-s − 2·11-s + 4·13-s − 2·17-s − 2·19-s + 4·23-s + 25-s − 6·29-s + 2·31-s + 35-s + 10·37-s + 10·41-s − 12·43-s − 8·47-s + 49-s + 2·55-s − 8·59-s − 2·61-s − 4·65-s + 12·67-s − 10·71-s + 4·73-s + 2·77-s − 12·83-s + 2·85-s − 2·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 0.603·11-s + 1.10·13-s − 0.485·17-s − 0.458·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.169·35-s + 1.64·37-s + 1.56·41-s − 1.82·43-s − 1.16·47-s + 1/7·49-s + 0.269·55-s − 1.04·59-s − 0.256·61-s − 0.496·65-s + 1.46·67-s − 1.18·71-s + 0.468·73-s + 0.227·77-s − 1.31·83-s + 0.216·85-s − 0.211·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 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
−7.979873266585939520965490547340, −7.15752052822925327458812773462, −6.42327068911189335007732550565, −5.80492916777264942063968462027, −4.86213707469792191146698739461, −4.11291598620944421354226506939, −3.31993382891113462575271751563, −2.50910852631925492238655521470, −1.28662298325762321994696944720, 0,
1.28662298325762321994696944720, 2.50910852631925492238655521470, 3.31993382891113462575271751563, 4.11291598620944421354226506939, 4.86213707469792191146698739461, 5.80492916777264942063968462027, 6.42327068911189335007732550565, 7.15752052822925327458812773462, 7.979873266585939520965490547340
