| L(s) = 1 | − 3-s − 3·5-s + 9-s − 11-s − 13-s + 3·15-s + 17-s + 2·19-s + 2·23-s + 4·25-s − 27-s − 10·29-s + 4·31-s + 33-s − 7·37-s + 39-s + 4·41-s + 43-s − 3·45-s + 8·47-s − 51-s − 9·53-s + 3·55-s − 2·57-s + 12·59-s − 10·61-s + 3·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.774·15-s + 0.242·17-s + 0.458·19-s + 0.417·23-s + 4/5·25-s − 0.192·27-s − 1.85·29-s + 0.718·31-s + 0.174·33-s − 1.15·37-s + 0.160·39-s + 0.624·41-s + 0.152·43-s − 0.447·45-s + 1.16·47-s − 0.140·51-s − 1.23·53-s + 0.404·55-s − 0.264·57-s + 1.56·59-s − 1.28·61-s + 0.372·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159936 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−13.46500118078462, −12.85372762905408, −12.50289798209089, −11.99827604837505, −11.70274509683013, −11.07691696458611, −10.91132163624756, −10.28420401110900, −9.702402769766150, −9.231235875797871, −8.656480274692972, −8.086040656697843, −7.595262295070024, −7.281492217772294, −6.861588105004409, −6.043827842905299, −5.649010346489427, −4.980853449085386, −4.646289619567194, −3.867774645012225, −3.616368650649344, −2.921055723231623, −2.200420164370021, −1.384584114115651, −0.6305900594787406, 0,
0.6305900594787406, 1.384584114115651, 2.200420164370021, 2.921055723231623, 3.616368650649344, 3.867774645012225, 4.646289619567194, 4.980853449085386, 5.649010346489427, 6.043827842905299, 6.861588105004409, 7.281492217772294, 7.595262295070024, 8.086040656697843, 8.656480274692972, 9.231235875797871, 9.702402769766150, 10.28420401110900, 10.91132163624756, 11.07691696458611, 11.70274509683013, 11.99827604837505, 12.50289798209089, 12.85372762905408, 13.46500118078462
