| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s + 12·5-s − 6·6-s + 4·7-s + 8·8-s + 9·9-s + 24·10-s + 8·11-s − 12·12-s + 24·13-s + 8·14-s − 36·15-s + 16·16-s + 62·17-s + 18·18-s + 48·20-s − 12·21-s + 16·22-s + 194·23-s − 24·24-s + 19·25-s + 48·26-s − 27·27-s + 16·28-s − 102·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 1.07·5-s − 0.408·6-s + 0.215·7-s + 0.353·8-s + 1/3·9-s + 0.758·10-s + 0.219·11-s − 0.288·12-s + 0.512·13-s + 0.152·14-s − 0.619·15-s + 1/4·16-s + 0.884·17-s + 0.235·18-s + 0.536·20-s − 0.124·21-s + 0.155·22-s + 1.75·23-s − 0.204·24-s + 0.151·25-s + 0.362·26-s − 0.192·27-s + 0.107·28-s − 0.653·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2166 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.519212189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.519212189\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| 19 | \( 1 \) |
| good | 5 | \( 1 - 12 T + p^{3} T^{2} \) |
| 7 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 8 T + p^{3} T^{2} \) |
| 13 | \( 1 - 24 T + p^{3} T^{2} \) |
| 17 | \( 1 - 62 T + p^{3} T^{2} \) |
| 23 | \( 1 - 194 T + p^{3} T^{2} \) |
| 29 | \( 1 + 102 T + p^{3} T^{2} \) |
| 31 | \( 1 + 18 T + p^{3} T^{2} \) |
| 37 | \( 1 - 8 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 134 T + p^{3} T^{2} \) |
| 43 | \( 1 + 60 T + p^{3} T^{2} \) |
| 47 | \( 1 + 226 T + p^{3} T^{2} \) |
| 53 | \( 1 - 362 T + p^{3} T^{2} \) |
| 59 | \( 1 - 316 T + p^{3} T^{2} \) |
| 61 | \( 1 - 134 T + p^{3} T^{2} \) |
| 67 | \( 1 - 240 T + p^{3} T^{2} \) |
| 71 | \( 1 - 800 T + p^{3} T^{2} \) |
| 73 | \( 1 + 578 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1078 T + p^{3} T^{2} \) |
| 83 | \( 1 - 940 T + p^{3} T^{2} \) |
| 89 | \( 1 + 170 T + p^{3} T^{2} \) |
| 97 | \( 1 + 206 T + p^{3} T^{2} \) |
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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
−8.796962912445007974421869022505, −7.75848887270464547687121760203, −6.87336527686762849175632651796, −6.20752663062646883886560093448, −5.48426596704685485132001467453, −4.97610523355829439642257054349, −3.88332540080844293326035645437, −2.91319899574009253268322756964, −1.78784999945951375471654594339, −0.954592338695060338480875266311,
0.954592338695060338480875266311, 1.78784999945951375471654594339, 2.91319899574009253268322756964, 3.88332540080844293326035645437, 4.97610523355829439642257054349, 5.48426596704685485132001467453, 6.20752663062646883886560093448, 6.87336527686762849175632651796, 7.75848887270464547687121760203, 8.796962912445007974421869022505
