| L(s) = 1 | − 3·7-s − 3·9-s + 3·11-s + 13-s − 17-s + 4·19-s + 2·23-s + 7·29-s + 5·31-s + 6·37-s + 4·41-s + 6·43-s − 13·47-s + 2·49-s + 9·53-s + 5·59-s + 13·61-s + 9·63-s − 5·67-s − 2·73-s − 9·77-s − 14·79-s + 9·81-s − 9·83-s + 4·89-s − 3·91-s − 2·97-s + ⋯ |
| L(s) = 1 | − 1.13·7-s − 9-s + 0.904·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 0.417·23-s + 1.29·29-s + 0.898·31-s + 0.986·37-s + 0.624·41-s + 0.914·43-s − 1.89·47-s + 2/7·49-s + 1.23·53-s + 0.650·59-s + 1.66·61-s + 1.13·63-s − 0.610·67-s − 0.234·73-s − 1.02·77-s − 1.57·79-s + 81-s − 0.987·83-s + 0.423·89-s − 0.314·91-s − 0.203·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.402029809\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.402029809\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.610687729109091860866987774642, −8.901915230199666122405908825588, −8.181746588611448836299713533434, −7.01020845057893015369300651529, −6.35194225795953866618109400854, −5.65084032863379860205720625866, −4.45369470832556879445489488255, −3.37692568846226166617918777384, −2.65679045954033804338141341614, −0.874633595172119022878525926783,
0.874633595172119022878525926783, 2.65679045954033804338141341614, 3.37692568846226166617918777384, 4.45369470832556879445489488255, 5.65084032863379860205720625866, 6.35194225795953866618109400854, 7.01020845057893015369300651529, 8.181746588611448836299713533434, 8.901915230199666122405908825588, 9.610687729109091860866987774642
