L(s) = 1 | − 2·3-s + 5-s − 4·7-s + 9-s + 11-s − 4·13-s − 2·15-s + 4·17-s + 8·21-s + 2·23-s + 25-s + 4·27-s − 6·29-s + 4·31-s − 2·33-s − 4·35-s − 10·37-s + 8·39-s − 2·41-s + 45-s − 2·47-s + 9·49-s − 8·51-s + 6·53-s + 55-s + 10·61-s − 4·63-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s − 0.516·15-s + 0.970·17-s + 1.74·21-s + 0.417·23-s + 1/5·25-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s − 0.676·35-s − 1.64·37-s + 1.28·39-s − 0.312·41-s + 0.149·45-s − 0.291·47-s + 9/7·49-s − 1.12·51-s + 0.824·53-s + 0.134·55-s + 1.28·61-s − 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 79420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 79420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 12 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.20347895955170, −13.71150024811520, −13.05546677359534, −12.71504724730262, −12.23860871225933, −11.84229183791662, −11.40085285911080, −10.62630190656247, −10.18007359994358, −9.927306656488401, −9.354511042267873, −8.876029077216147, −8.161668690586485, −7.348794744605790, −6.822164800426292, −6.664193419797230, −5.816225907064064, −5.562087043133111, −5.086101441089338, −4.342399282728957, −3.565004923116184, −3.064113283515367, −2.427743029472347, −1.524669076506479, −0.6548142813524517, 0,
0.6548142813524517, 1.524669076506479, 2.427743029472347, 3.064113283515367, 3.565004923116184, 4.342399282728957, 5.086101441089338, 5.562087043133111, 5.816225907064064, 6.664193419797230, 6.822164800426292, 7.348794744605790, 8.161668690586485, 8.876029077216147, 9.354511042267873, 9.927306656488401, 10.18007359994358, 10.62630190656247, 11.40085285911080, 11.84229183791662, 12.23860871225933, 12.71504724730262, 13.05546677359534, 13.71150024811520, 14.20347895955170