L(s) = 1 | − 2·3-s − 3·5-s + 7-s + 9-s + 6·15-s − 6·17-s + 7·19-s − 2·21-s − 3·23-s + 4·25-s + 4·27-s + 9·29-s + 5·31-s − 3·35-s + 2·37-s + 6·41-s − 43-s − 3·45-s + 3·47-s + 49-s + 12·51-s + 9·53-s − 14·57-s + 10·61-s + 63-s − 14·67-s + 6·69-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 1.54·15-s − 1.45·17-s + 1.60·19-s − 0.436·21-s − 0.625·23-s + 4/5·25-s + 0.769·27-s + 1.67·29-s + 0.898·31-s − 0.507·35-s + 0.328·37-s + 0.937·41-s − 0.152·43-s − 0.447·45-s + 0.437·47-s + 1/7·49-s + 1.68·51-s + 1.23·53-s − 1.85·57-s + 1.28·61-s + 0.125·63-s − 1.71·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−14.29410559654615, −13.75281970024720, −13.34451844197946, −12.52667398336632, −12.09589855459471, −11.72496536637971, −11.45684694979608, −11.03968759510621, −10.34367125359209, −10.06471934173203, −9.150128918073006, −8.623745242399577, −8.186373243893771, −7.581673969607055, −7.093627645390466, −6.597580227940029, −5.968620547896759, −5.439706750582827, −4.771893158377331, −4.354922239251239, −3.951981838555965, −2.950508031347986, −2.561080392942125, −1.324836742072009, −0.7366024105076901, 0,
0.7366024105076901, 1.324836742072009, 2.561080392942125, 2.950508031347986, 3.951981838555965, 4.354922239251239, 4.771893158377331, 5.439706750582827, 5.968620547896759, 6.597580227940029, 7.093627645390466, 7.581673969607055, 8.186373243893771, 8.623745242399577, 9.150128918073006, 10.06471934173203, 10.34367125359209, 11.03968759510621, 11.45684694979608, 11.72496536637971, 12.09589855459471, 12.52667398336632, 13.34451844197946, 13.75281970024720, 14.29410559654615