L(s) = 1 | − 5-s − 11-s − 5·13-s + 17-s + 8·19-s − 6·23-s − 4·25-s + 4·29-s − 2·31-s + 3·37-s + 6·41-s − 13·43-s − 2·47-s − 5·53-s + 55-s + 4·59-s + 2·61-s + 5·65-s + 5·67-s + 14·71-s + 11·73-s − 79-s + 15·83-s − 85-s − 5·89-s − 8·95-s − 97-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.301·11-s − 1.38·13-s + 0.242·17-s + 1.83·19-s − 1.25·23-s − 4/5·25-s + 0.742·29-s − 0.359·31-s + 0.493·37-s + 0.937·41-s − 1.98·43-s − 0.291·47-s − 0.686·53-s + 0.134·55-s + 0.520·59-s + 0.256·61-s + 0.620·65-s + 0.610·67-s + 1.66·71-s + 1.28·73-s − 0.112·79-s + 1.64·83-s − 0.108·85-s − 0.529·89-s − 0.820·95-s − 0.101·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 13 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 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
−13.93256607221367, −13.38754310389129, −12.68959698175236, −12.35014874617273, −11.77115753264652, −11.61100157683020, −11.01033389596214, −10.16546864662823, −9.970701802551994, −9.523226501595985, −9.044386618140584, −8.012425367102430, −7.952025683791998, −7.592271936524474, −6.795580025332651, −6.461842704087489, −5.599685920002761, −5.161412768449826, −4.836827014928980, −3.939318622517445, −3.613484407522237, −2.839665990391381, −2.351586666068080, −1.613074231374149, −0.7419332396553261, 0,
0.7419332396553261, 1.613074231374149, 2.351586666068080, 2.839665990391381, 3.613484407522237, 3.939318622517445, 4.836827014928980, 5.161412768449826, 5.599685920002761, 6.461842704087489, 6.795580025332651, 7.592271936524474, 7.952025683791998, 8.012425367102430, 9.044386618140584, 9.523226501595985, 9.970701802551994, 10.16546864662823, 11.01033389596214, 11.61100157683020, 11.77115753264652, 12.35014874617273, 12.68959698175236, 13.38754310389129, 13.93256607221367