| L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s + 2·5-s + 6·6-s − 7·7-s + 8·8-s + 9·9-s + 4·10-s − 8·11-s + 12·12-s − 42·13-s − 14·14-s + 6·15-s + 16·16-s − 2·17-s + 18·18-s − 124·19-s + 8·20-s − 21·21-s − 16·22-s + 76·23-s + 24·24-s − 121·25-s − 84·26-s + 27·27-s − 28·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.178·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.126·10-s − 0.219·11-s + 0.288·12-s − 0.896·13-s − 0.267·14-s + 0.103·15-s + 1/4·16-s − 0.0285·17-s + 0.235·18-s − 1.49·19-s + 0.0894·20-s − 0.218·21-s − 0.155·22-s + 0.689·23-s + 0.204·24-s − 0.967·25-s − 0.633·26-s + 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.122213491\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.122213491\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 8 T + p^{3} T^{2} \) |
| 13 | \( 1 + 42 T + p^{3} T^{2} \) |
| 17 | \( 1 + 2 T + p^{3} T^{2} \) |
| 19 | \( 1 + 124 T + p^{3} T^{2} \) |
| 23 | \( 1 - 76 T + p^{3} T^{2} \) |
| 29 | \( 1 - 254 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 398 T + p^{3} T^{2} \) |
| 41 | \( 1 - 462 T + p^{3} T^{2} \) |
| 43 | \( 1 - 212 T + p^{3} T^{2} \) |
| 47 | \( 1 + 264 T + p^{3} T^{2} \) |
| 53 | \( 1 + 162 T + p^{3} T^{2} \) |
| 59 | \( 1 + 772 T + p^{3} T^{2} \) |
| 61 | \( 1 - 30 T + p^{3} T^{2} \) |
| 67 | \( 1 + 764 T + p^{3} T^{2} \) |
| 71 | \( 1 + 236 T + p^{3} T^{2} \) |
| 73 | \( 1 - 418 T + p^{3} T^{2} \) |
| 79 | \( 1 - 552 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1036 T + p^{3} T^{2} \) |
| 89 | \( 1 - 30 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1190 T + p^{3} T^{2} \) |
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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
−15.24907888356959080078119457131, −14.38807398338182833411677578000, −13.21770227343104200983664078423, −12.34649018301336546275646549967, −10.73751196562222595962090365806, −9.413712194209449682264304021629, −7.78628074367903158375486086399, −6.29037507773754274652042152738, −4.46044875094035441087699494163, −2.60588720017564028889637504318,
2.60588720017564028889637504318, 4.46044875094035441087699494163, 6.29037507773754274652042152738, 7.78628074367903158375486086399, 9.413712194209449682264304021629, 10.73751196562222595962090365806, 12.34649018301336546275646549967, 13.21770227343104200983664078423, 14.38807398338182833411677578000, 15.24907888356959080078119457131
