| L(s) = 1 | − 2.30·3-s − 3·5-s − 3.30·7-s + 2.30·9-s + 6.30·13-s + 6.90·15-s − 1.30·17-s − 2·19-s + 7.60·21-s − 1.30·23-s + 4·25-s + 1.60·27-s + 0.394·29-s + 0.605·31-s + 9.90·35-s + 7.60·37-s − 14.5·39-s + 8.21·41-s − 2.39·43-s − 6.90·45-s + 5.60·47-s + 3.90·49-s + 3·51-s + 2.60·53-s + 4.60·57-s + 13.8·59-s − 14.8·61-s + ⋯ |
| L(s) = 1 | − 1.32·3-s − 1.34·5-s − 1.24·7-s + 0.767·9-s + 1.74·13-s + 1.78·15-s − 0.315·17-s − 0.458·19-s + 1.65·21-s − 0.271·23-s + 0.800·25-s + 0.308·27-s + 0.0732·29-s + 0.108·31-s + 1.67·35-s + 1.25·37-s − 2.32·39-s + 1.28·41-s − 0.365·43-s − 1.02·45-s + 0.817·47-s + 0.558·49-s + 0.420·51-s + 0.357·53-s + 0.610·57-s + 1.79·59-s − 1.89·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 + 3T + 5T^{2} \) |
| 7 | \( 1 + 3.30T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 6.30T + 13T^{2} \) |
| 17 | \( 1 + 1.30T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 1.30T + 23T^{2} \) |
| 29 | \( 1 - 0.394T + 29T^{2} \) |
| 31 | \( 1 - 0.605T + 31T^{2} \) |
| 37 | \( 1 - 7.60T + 37T^{2} \) |
| 41 | \( 1 - 8.21T + 41T^{2} \) |
| 43 | \( 1 + 2.39T + 43T^{2} \) |
| 47 | \( 1 - 5.60T + 47T^{2} \) |
| 53 | \( 1 - 2.60T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 - 6.21T + 67T^{2} \) |
| 71 | \( 1 - 6.90T + 71T^{2} \) |
| 73 | \( 1 + 13.9T + 73T^{2} \) |
| 79 | \( 1 + 13.6T + 79T^{2} \) |
| 83 | \( 1 + 5.60T + 83T^{2} \) |
| 89 | \( 1 - 7.81T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−8.457337631635048081208002565346, −7.61421097594578418356217522499, −6.69209985320981824150921740501, −6.22045255710074564239205228761, −5.57887927496984136866493809621, −4.30113307120570612629483582965, −3.89486290341990691446118793637, −2.87154547946237865816102667459, −0.978043654229384867749334486503, 0,
0.978043654229384867749334486503, 2.87154547946237865816102667459, 3.89486290341990691446118793637, 4.30113307120570612629483582965, 5.57887927496984136866493809621, 6.22045255710074564239205228761, 6.69209985320981824150921740501, 7.61421097594578418356217522499, 8.457337631635048081208002565346
