| L(s) = 1 | + 3-s − 7-s + 9-s + 3·11-s − 13-s + 5·17-s + 2·19-s − 21-s + 3·23-s + 27-s − 4·31-s + 3·33-s + 37-s − 39-s + 9·41-s + 2·43-s − 8·47-s − 6·49-s + 5·51-s − 53-s + 2·57-s − 4·59-s + 3·61-s − 63-s − 16·67-s + 3·69-s − 15·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s + 1.21·17-s + 0.458·19-s − 0.218·21-s + 0.625·23-s + 0.192·27-s − 0.718·31-s + 0.522·33-s + 0.164·37-s − 0.160·39-s + 1.40·41-s + 0.304·43-s − 1.16·47-s − 6/7·49-s + 0.700·51-s − 0.137·53-s + 0.264·57-s − 0.520·59-s + 0.384·61-s − 0.125·63-s − 1.95·67-s + 0.361·69-s − 1.78·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 62400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.51576162828456, −14.22467541755265, −13.48994048195133, −13.02258492100854, −12.55457175009199, −12.07188307633971, −11.51505820854671, −11.02486169302105, −10.30448616064149, −9.852677087756527, −9.294708296505690, −9.084159758271649, −8.319509583998066, −7.736278997504875, −7.322738340159937, −6.770868992378661, −6.104584146685293, −5.605826591803273, −4.896837339473521, −4.242607945921959, −3.660634210275701, −3.062915520161001, −2.630458328938117, −1.530205763314255, −1.188166299730477, 0,
1.188166299730477, 1.530205763314255, 2.630458328938117, 3.062915520161001, 3.660634210275701, 4.242607945921959, 4.896837339473521, 5.605826591803273, 6.104584146685293, 6.770868992378661, 7.322738340159937, 7.736278997504875, 8.319509583998066, 9.084159758271649, 9.294708296505690, 9.852677087756527, 10.30448616064149, 11.02486169302105, 11.51505820854671, 12.07188307633971, 12.55457175009199, 13.02258492100854, 13.48994048195133, 14.22467541755265, 14.51576162828456
