| L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s − 4·7-s + 8-s + 9-s + 6·11-s − 2·12-s + 2·13-s − 4·14-s + 16-s − 17-s + 18-s − 4·19-s + 8·21-s + 6·22-s − 2·24-s − 5·25-s + 2·26-s + 4·27-s − 4·28-s − 4·31-s + 32-s − 12·33-s − 34-s + 36-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s − 1.51·7-s + 0.353·8-s + 1/3·9-s + 1.80·11-s − 0.577·12-s + 0.554·13-s − 1.06·14-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 0.917·19-s + 1.74·21-s + 1.27·22-s − 0.408·24-s − 25-s + 0.392·26-s + 0.769·27-s − 0.755·28-s − 0.718·31-s + 0.176·32-s − 2.08·33-s − 0.171·34-s + 1/6·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7492772210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7492772210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.63179484427491344136926072682, −15.73128059955905922020314753971, −14.21978494242446158082804784774, −12.85956463289156748627394436952, −11.95834417367800404279447796077, −10.84894668716717132123408330664, −9.286071988264369821451329617999, −6.65330731261113455462227374582, −5.99911855171913780844071238816, −3.90229547122613769308947697962,
3.90229547122613769308947697962, 5.99911855171913780844071238816, 6.65330731261113455462227374582, 9.286071988264369821451329617999, 10.84894668716717132123408330664, 11.95834417367800404279447796077, 12.85956463289156748627394436952, 14.21978494242446158082804784774, 15.73128059955905922020314753971, 16.63179484427491344136926072682
