L(s) = 1 | + 5·11-s + 2·13-s + 6·17-s + 2·19-s − 5·23-s + 5·29-s + 4·31-s + 37-s − 12·41-s + 5·43-s + 2·47-s − 14·53-s + 2·59-s − 5·67-s + 9·71-s − 10·73-s + 11·79-s + 16·83-s − 14·89-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 1.50·11-s + 0.554·13-s + 1.45·17-s + 0.458·19-s − 1.04·23-s + 0.928·29-s + 0.718·31-s + 0.164·37-s − 1.87·41-s + 0.762·43-s + 0.291·47-s − 1.92·53-s + 0.260·59-s − 0.610·67-s + 1.06·71-s − 1.17·73-s + 1.23·79-s + 1.75·83-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 88200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.451075850\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.451075850\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 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
−13.92728537838154, −13.62707059034324, −12.81946008760880, −12.21086313953925, −11.97644525929100, −11.60548413472942, −10.92516529158320, −10.35964677253864, −9.827190017462809, −9.505986576026217, −8.888750158391086, −8.263185097400190, −7.974872287032737, −7.260564400166713, −6.677535814447108, −6.170394620790393, −5.830838158732695, −5.007512184788632, −4.523097394197315, −3.735962241975457, −3.465827749450895, −2.774221747074062, −1.794548761708685, −1.299853369812115, −0.6488482981373373,
0.6488482981373373, 1.299853369812115, 1.794548761708685, 2.774221747074062, 3.465827749450895, 3.735962241975457, 4.523097394197315, 5.007512184788632, 5.830838158732695, 6.170394620790393, 6.677535814447108, 7.260564400166713, 7.974872287032737, 8.263185097400190, 8.888750158391086, 9.505986576026217, 9.827190017462809, 10.35964677253864, 10.92516529158320, 11.60548413472942, 11.97644525929100, 12.21086313953925, 12.81946008760880, 13.62707059034324, 13.92728537838154