| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s + 5·11-s − 5·13-s + 2·14-s + 16-s + 3·17-s − 5·19-s − 5·22-s + 23-s + 5·26-s − 2·28-s + 5·29-s − 7·31-s − 32-s − 3·34-s + 5·37-s + 5·38-s + 10·41-s + 4·43-s + 5·44-s − 46-s − 2·47-s − 3·49-s − 5·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s + 1.50·11-s − 1.38·13-s + 0.534·14-s + 1/4·16-s + 0.727·17-s − 1.14·19-s − 1.06·22-s + 0.208·23-s + 0.980·26-s − 0.377·28-s + 0.928·29-s − 1.25·31-s − 0.176·32-s − 0.514·34-s + 0.821·37-s + 0.811·38-s + 1.56·41-s + 0.609·43-s + 0.753·44-s − 0.147·46-s − 0.291·47-s − 3/7·49-s − 0.693·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.48542773225344, −14.04102753487079, −12.96710038998023, −12.87104984656534, −12.25409384990132, −11.81079071762419, −11.32907988696654, −10.73795706772921, −10.04220313096514, −9.830610802135118, −9.274575185882691, −8.853093636369514, −8.313831782321638, −7.608730276658888, −7.015299849921904, −6.834218015232362, −6.049838330051495, −5.699928355031287, −4.838561537611506, −4.074029904764120, −3.777948361503697, −2.737738157849756, −2.484238586720621, −1.530621841605050, −0.8460332332197011, 0,
0.8460332332197011, 1.530621841605050, 2.484238586720621, 2.737738157849756, 3.777948361503697, 4.074029904764120, 4.838561537611506, 5.699928355031287, 6.049838330051495, 6.834218015232362, 7.015299849921904, 7.608730276658888, 8.313831782321638, 8.853093636369514, 9.274575185882691, 9.830610802135118, 10.04220313096514, 10.73795706772921, 11.32907988696654, 11.81079071762419, 12.25409384990132, 12.87104984656534, 12.96710038998023, 14.04102753487079, 14.48542773225344
