| 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
| \(L(1)\) |
\(\approx\) |
\(2.845748901\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.845748901\) |
| \(L(\frac{3}{2})\) |
|
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
−12.87029382394650, −12.78852021954831, −12.18447858864307, −11.67103979275523, −11.15100146193906, −10.88824573128090, −10.37979345707361, −9.787015824428167, −9.491024959775370, −8.602421662480182, −8.301143189413669, −7.811379934827961, −7.632185595751309, −6.935608262547461, −6.264405871439428, −5.562228968714217, −5.463686452659912, −4.593131289428399, −4.413779106397932, −3.796352140054897, −2.873792037140478, −2.360175687550103, −2.009658374480869, −1.141980455060425, −0.5112275381656092,
0.5112275381656092, 1.141980455060425, 2.009658374480869, 2.360175687550103, 2.873792037140478, 3.796352140054897, 4.413779106397932, 4.593131289428399, 5.463686452659912, 5.562228968714217, 6.264405871439428, 6.935608262547461, 7.632185595751309, 7.811379934827961, 8.301143189413669, 8.602421662480182, 9.491024959775370, 9.787015824428167, 10.37979345707361, 10.88824573128090, 11.15100146193906, 11.67103979275523, 12.18447858864307, 12.78852021954831, 12.87029382394650
