| L(s) = 1 | − 5-s − 2·7-s − 11-s + 13-s − 6·17-s + 4·19-s + 25-s − 2·31-s + 2·35-s − 4·37-s − 6·41-s − 8·43-s − 3·49-s − 12·53-s + 55-s − 10·61-s − 65-s − 8·67-s − 10·73-s + 2·77-s − 8·79-s + 6·85-s − 6·89-s − 2·91-s − 4·95-s − 16·97-s + 101-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s − 0.301·11-s + 0.277·13-s − 1.45·17-s + 0.917·19-s + 1/5·25-s − 0.359·31-s + 0.338·35-s − 0.657·37-s − 0.937·41-s − 1.21·43-s − 3/7·49-s − 1.64·53-s + 0.134·55-s − 1.28·61-s − 0.124·65-s − 0.977·67-s − 1.17·73-s + 0.227·77-s − 0.900·79-s + 0.650·85-s − 0.635·89-s − 0.209·91-s − 0.410·95-s − 1.62·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−14.09636025354956, −13.58428778711057, −13.33635591682134, −12.72362845680040, −12.34990172550416, −11.70433888451230, −11.31532740452519, −10.84123012588849, −10.29209628996857, −9.745474982525764, −9.323224825724310, −8.678877467109244, −8.383709552661879, −7.669527615233148, −7.126294779210399, −6.738666241509493, −6.166085233282025, −5.635703493978233, −4.852055327264426, −4.550754189507459, −3.754378086886929, −3.192359750590333, −2.851181268461256, −1.881862626488060, −1.326793471954305, 0, 0,
1.326793471954305, 1.881862626488060, 2.851181268461256, 3.192359750590333, 3.754378086886929, 4.550754189507459, 4.852055327264426, 5.635703493978233, 6.166085233282025, 6.738666241509493, 7.126294779210399, 7.669527615233148, 8.383709552661879, 8.678877467109244, 9.323224825724310, 9.745474982525764, 10.29209628996857, 10.84123012588849, 11.31532740452519, 11.70433888451230, 12.34990172550416, 12.72362845680040, 13.33635591682134, 13.58428778711057, 14.09636025354956
