| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 11-s + 13-s + 4·14-s + 16-s − 6·19-s − 20-s + 22-s − 4·23-s + 25-s + 26-s + 4·28-s + 10·29-s − 4·31-s + 32-s − 4·35-s − 4·37-s − 6·38-s − 40-s + 6·41-s + 4·43-s + 44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 0.301·11-s + 0.277·13-s + 1.06·14-s + 1/4·16-s − 1.37·19-s − 0.223·20-s + 0.213·22-s − 0.834·23-s + 1/5·25-s + 0.196·26-s + 0.755·28-s + 1.85·29-s − 0.718·31-s + 0.176·32-s − 0.676·35-s − 0.657·37-s − 0.973·38-s − 0.158·40-s + 0.937·41-s + 0.609·43-s + 0.150·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.048689880\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.048689880\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−16.13031170385184, −15.59248695395118, −15.06593509477649, −14.43809304208571, −14.20761845638499, −13.61383294565548, −12.72416760550260, −12.34881646824468, −11.74862678733093, −11.16323298604993, −10.79417240504792, −10.18019205239081, −9.222367127862637, −8.398374470065100, −8.154857206979183, −7.498492260430296, −6.604837034525427, −6.215877629259874, −5.201679217932833, −4.782860630359450, −4.104897346637258, −3.578557164037082, −2.406523151410575, −1.857213713438661, −0.8286984147821988,
0.8286984147821988, 1.857213713438661, 2.406523151410575, 3.578557164037082, 4.104897346637258, 4.782860630359450, 5.201679217932833, 6.215877629259874, 6.604837034525427, 7.498492260430296, 8.154857206979183, 8.398374470065100, 9.222367127862637, 10.18019205239081, 10.79417240504792, 11.16323298604993, 11.74862678733093, 12.34881646824468, 12.72416760550260, 13.61383294565548, 14.20761845638499, 14.43809304208571, 15.06593509477649, 15.59248695395118, 16.13031170385184
