L(s) = 1 | + 3-s + 2·5-s + 4·7-s + 9-s − 11-s + 4·13-s + 2·15-s − 17-s − 8·19-s + 4·21-s − 25-s + 27-s − 10·31-s − 33-s + 8·35-s − 8·37-s + 4·39-s − 10·41-s − 8·43-s + 2·45-s − 10·47-s + 9·49-s − 51-s + 12·53-s − 2·55-s − 8·57-s − 8·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.516·15-s − 0.242·17-s − 1.83·19-s + 0.872·21-s − 1/5·25-s + 0.192·27-s − 1.79·31-s − 0.174·33-s + 1.35·35-s − 1.31·37-s + 0.640·39-s − 1.56·41-s − 1.21·43-s + 0.298·45-s − 1.45·47-s + 9/7·49-s − 0.140·51-s + 1.64·53-s − 0.269·55-s − 1.05·57-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.14136581541832, −14.62782684392030, −14.12482620283217, −13.64770356551737, −13.18673772735002, −12.80647497715180, −11.97945795662443, −11.41087966489883, −10.80500049165534, −10.53588073718648, −9.915450912403711, −9.135871198589672, −8.645750356221726, −8.294734273423989, −7.869050209530566, −6.830850881396263, −6.646529630808876, −5.634789928049777, −5.316131875945708, −4.602709664491860, −3.924604279183733, −3.348904193204165, −2.266537759969740, −1.798029850461521, −1.496112185367415, 0,
1.496112185367415, 1.798029850461521, 2.266537759969740, 3.348904193204165, 3.924604279183733, 4.602709664491860, 5.316131875945708, 5.634789928049777, 6.646529630808876, 6.830850881396263, 7.869050209530566, 8.294734273423989, 8.645750356221726, 9.135871198589672, 9.915450912403711, 10.53588073718648, 10.80500049165534, 11.41087966489883, 11.97945795662443, 12.80647497715180, 13.18673772735002, 13.64770356551737, 14.12482620283217, 14.62782684392030, 15.14136581541832