L(s) = 1 | + i·2-s − 3i·3-s − 4-s + 3·6-s − i·8-s − 6·9-s − 5·11-s + 3i·12-s − 6i·13-s + 16-s + i·17-s − 6i·18-s − 3·19-s − 5i·22-s − 3·24-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 1.73i·3-s − 0.5·4-s + 1.22·6-s − 0.353i·8-s − 2·9-s − 1.50·11-s + 0.866i·12-s − 1.66i·13-s + 0.250·16-s + 0.242i·17-s − 1.41i·18-s − 0.688·19-s − 1.06i·22-s − 0.612·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 3iT - 3T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 - iT - 17T^{2} \) |
| 19 | \( 1 + 3T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 11T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + 4iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 9iT - 67T^{2} \) |
| 71 | \( 1 + 10T + 71T^{2} \) |
| 73 | \( 1 + 7iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 11iT - 83T^{2} \) |
| 89 | \( 1 + 11T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−8.240062220808379565180363808354, −7.75221844997496956988729162755, −6.88813602571735589824037104794, −6.27776705825829606733848795820, −5.51659758232396871381930395763, −4.83535015245104865470929570050, −3.15619455694130266348025053715, −2.49071683744559344059936404983, −1.11675087360072396431518897735, 0,
2.12181144244860667873958093578, 2.97191928692346512057964156740, 3.91841885518190209386700625795, 4.60487297833844137013602898135, 5.11002405248045217891381302415, 6.07165830844898030545968820545, 7.22972871033631525050809869205, 8.480645719412381995950143571902, 8.792311736872499710709956111953