| L(s) = 1 | − 4·11-s + 17-s − 2·19-s − 8·23-s − 5·25-s − 6·29-s + 4·31-s − 8·41-s − 8·43-s − 2·47-s − 7·49-s − 10·53-s − 6·59-s + 6·61-s + 14·67-s + 4·73-s − 8·79-s − 6·83-s + 14·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.20·11-s + 0.242·17-s − 0.458·19-s − 1.66·23-s − 25-s − 1.11·29-s + 0.718·31-s − 1.24·41-s − 1.21·43-s − 0.291·47-s − 49-s − 1.37·53-s − 0.781·59-s + 0.768·61-s + 1.71·67-s + 0.468·73-s − 0.900·79-s − 0.658·83-s + 1.48·89-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103428 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.14036941826513, −13.71970868637730, −13.19485105819244, −12.81180566553131, −12.31594436748843, −11.67340361099245, −11.37634541891120, −10.77112316494211, −10.12552900937330, −9.889496271713330, −9.483697895450507, −8.492614530672672, −8.278822631416714, −7.787447998892146, −7.335430261215389, −6.527369270493040, −6.149553794555361, −5.570541149369850, −4.980364700963020, −4.566402252990148, −3.627236460884990, −3.465844167908392, −2.465436294771985, −2.043128339928795, −1.382634312991862, 0, 0,
1.382634312991862, 2.043128339928795, 2.465436294771985, 3.465844167908392, 3.627236460884990, 4.566402252990148, 4.980364700963020, 5.570541149369850, 6.149553794555361, 6.527369270493040, 7.335430261215389, 7.787447998892146, 8.278822631416714, 8.492614530672672, 9.483697895450507, 9.889496271713330, 10.12552900937330, 10.77112316494211, 11.37634541891120, 11.67340361099245, 12.31594436748843, 12.81180566553131, 13.19485105819244, 13.71970868637730, 14.14036941826513
