| L(s) = 1 | + i·3-s − i·7-s − 9-s + 2·11-s + 6i·13-s − 4i·17-s + 4·19-s + 21-s − 2i·23-s − i·27-s + 2·29-s + 2i·33-s + 2i·37-s − 6·39-s + 4i·43-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.377i·7-s − 0.333·9-s + 0.603·11-s + 1.66i·13-s − 0.970i·17-s + 0.917·19-s + 0.218·21-s − 0.417i·23-s − 0.192i·27-s + 0.371·29-s + 0.348i·33-s + 0.328i·37-s − 0.960·39-s + 0.609i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.758261932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.758261932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 2iT - 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 12iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.410842450543863437174383112549, −8.660605296538704575010713847983, −7.63610894803344717515245711160, −6.85739970011255391754245817532, −6.19616678962807287090231847464, −5.01307876736874911345931860191, −4.40875461229187015404561513212, −3.57901358921193336593023507932, −2.49116600547936501545887035581, −1.13047142605742605658667390980,
0.74047661061743712882455587303, 1.94162255760133997711134344667, 3.07819373238062514447643808718, 3.85556634225837624608114609035, 5.28631737703050010564251201895, 5.69515295677129392827215795130, 6.66879814880379010046535380851, 7.42694250125682410306923920463, 8.259656528500359648891736466763, 8.718276897808999994117697582815
