L(s) = 1 | − 2·7-s + 13-s − 2·17-s + 2·19-s + 8·23-s − 5·25-s − 6·29-s − 2·31-s − 6·37-s − 4·43-s + 8·47-s − 3·49-s + 6·53-s + 4·59-s + 2·61-s − 2·67-s + 4·71-s − 2·73-s − 12·79-s − 12·83-s − 12·89-s − 2·91-s − 18·97-s + 2·101-s − 12·103-s − 4·107-s − 2·109-s + ⋯ |
L(s) = 1 | − 0.755·7-s + 0.277·13-s − 0.485·17-s + 0.458·19-s + 1.66·23-s − 25-s − 1.11·29-s − 0.359·31-s − 0.986·37-s − 0.609·43-s + 1.16·47-s − 3/7·49-s + 0.824·53-s + 0.520·59-s + 0.256·61-s − 0.244·67-s + 0.474·71-s − 0.234·73-s − 1.35·79-s − 1.31·83-s − 1.27·89-s − 0.209·91-s − 1.82·97-s + 0.199·101-s − 1.18·103-s − 0.386·107-s − 0.191·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.218923880935360823370612191861, −7.16094868344855288450608318340, −6.88822552792408483108820982071, −5.80939474641690293006845652285, −5.29345936355717644120860948526, −4.17671090112527503871467157185, −3.45278696937777469414154368862, −2.59732209903582349912141546878, −1.41703594163353409210687841436, 0,
1.41703594163353409210687841436, 2.59732209903582349912141546878, 3.45278696937777469414154368862, 4.17671090112527503871467157185, 5.29345936355717644120860948526, 5.80939474641690293006845652285, 6.88822552792408483108820982071, 7.16094868344855288450608318340, 8.218923880935360823370612191861