| L(s) = 1 | − 4·7-s − 4·11-s − 13-s − 8·23-s − 8·29-s + 4·31-s + 6·37-s − 12·41-s − 8·43-s − 4·47-s + 9·49-s + 4·59-s + 2·61-s − 8·67-s + 4·71-s + 10·73-s + 16·77-s − 4·79-s − 12·83-s + 12·89-s + 4·91-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s − 0.277·13-s − 1.66·23-s − 1.48·29-s + 0.718·31-s + 0.986·37-s − 1.87·41-s − 1.21·43-s − 0.583·47-s + 9/7·49-s + 0.520·59-s + 0.256·61-s − 0.977·67-s + 0.474·71-s + 1.17·73-s + 1.82·77-s − 0.450·79-s − 1.31·83-s + 1.27·89-s + 0.419·91-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.21480595417329, −13.02481246030065, −12.51340615527087, −11.97328536233985, −11.56610079339296, −11.01028033209367, −10.26467691831439, −10.00248756030727, −9.832249866476828, −9.168948204578630, −8.580412276624609, −8.020547822891057, −7.696785149898302, −7.046610182094693, −6.522346232473321, −6.173504898355766, −5.519255690463935, −5.182617445092504, −4.423755889225300, −3.831208917386786, −3.311711419800548, −2.852696512960852, −2.204706305291929, −1.667457413709754, −0.4886908325364618, 0,
0.4886908325364618, 1.667457413709754, 2.204706305291929, 2.852696512960852, 3.311711419800548, 3.831208917386786, 4.423755889225300, 5.182617445092504, 5.519255690463935, 6.173504898355766, 6.522346232473321, 7.046610182094693, 7.696785149898302, 8.020547822891057, 8.580412276624609, 9.168948204578630, 9.832249866476828, 10.00248756030727, 10.26467691831439, 11.01028033209367, 11.56610079339296, 11.97328536233985, 12.51340615527087, 13.02481246030065, 13.21480595417329
