| L(s) = 1 | + (−0.5 + 2.17i)5-s − 3·7-s + 3·9-s + 4.35i·11-s − 4.35i·17-s + 4.35i·19-s − 4·23-s + (−4.50 − 2.17i)25-s + (1.5 − 6.53i)35-s + 43-s + (−1.5 + 6.53i)45-s − 13·47-s + 2·49-s + (−9.50 − 2.17i)55-s − 15·61-s + ⋯ |
| L(s) = 1 | + (−0.223 + 0.974i)5-s − 1.13·7-s + 9-s + 1.31i·11-s − 1.05i·17-s + 0.999i·19-s − 0.834·23-s + (−0.900 − 0.435i)25-s + (0.253 − 1.10i)35-s + 0.152·43-s + (−0.223 + 0.974i)45-s − 1.89·47-s + 0.285·49-s + (−1.28 − 0.293i)55-s − 1.92·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6711353768\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6711353768\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 - 2.17i)T \) |
| 19 | \( 1 - 4.35iT \) |
| good | 3 | \( 1 - 3T^{2} \) |
| 7 | \( 1 + 3T + 7T^{2} \) |
| 11 | \( 1 - 4.35iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.35iT - 17T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - T + 43T^{2} \) |
| 47 | \( 1 + 13T + 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 15T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 13.0iT - 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−9.943609902467584176150811570075, −9.407916954350084663762479816758, −7.995338139421489789655875907210, −7.26413045999532728984837754399, −6.76825466348911960981890906694, −5.98266824490886582893746105780, −4.67806006199149807135392483161, −3.83773599499473503083648746231, −2.93465351046656711873632424033, −1.81071199655440048009678262106,
0.25981117244050307337377077184, 1.57140202612128573553902789592, 3.14814889550664083957276970171, 3.94930041692814403084306975469, 4.83249911014058155297738801759, 5.95551863384690287101142259616, 6.49422194744953233670538523205, 7.59596642192616466280743094962, 8.382939353457827107705217795116, 9.103799287178670491724369060948
