| L(s) = 1 | + 3.41·7-s − 2.58·11-s − 3.41i·13-s − 1.17·17-s − 4.82i·23-s + 6i·29-s + 6.48i·31-s + 9.07i·37-s − 11.0i·41-s + 6.82·43-s − 5.65i·47-s + 4.65·49-s + 1.17·53-s + 6.58·59-s + 12.8·61-s + ⋯ |
| L(s) = 1 | + 1.29·7-s − 0.779·11-s − 0.946i·13-s − 0.284·17-s − 1.00i·23-s + 1.11i·29-s + 1.16i·31-s + 1.49i·37-s − 1.72i·41-s + 1.04·43-s − 0.825i·47-s + 0.665·49-s + 0.160·53-s + 0.857·59-s + 1.64·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.064729504\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.064729504\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 + 2.58T + 11T^{2} \) |
| 13 | \( 1 + 3.41iT - 13T^{2} \) |
| 17 | \( 1 + 1.17T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 4.82iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 6.48iT - 31T^{2} \) |
| 37 | \( 1 - 9.07iT - 37T^{2} \) |
| 41 | \( 1 + 11.0iT - 41T^{2} \) |
| 43 | \( 1 - 6.82T + 43T^{2} \) |
| 47 | \( 1 + 5.65iT - 47T^{2} \) |
| 53 | \( 1 - 1.17T + 53T^{2} \) |
| 59 | \( 1 - 6.58T + 59T^{2} \) |
| 61 | \( 1 - 12.8T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 5.65T + 71T^{2} \) |
| 73 | \( 1 + 10.4iT - 73T^{2} \) |
| 79 | \( 1 + 14.4iT - 79T^{2} \) |
| 83 | \( 1 + 9.17iT - 83T^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + 2.48iT - 97T^{2} \) |
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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
−7.84583443366465581954562175618, −7.24418454235944004049750934175, −6.49208334189484991998395503453, −5.40229658067524275772792384544, −5.14477394395156324200920275953, −4.38161078184177786715753935386, −3.37383893043293943385982784151, −2.53912892819140806009814650276, −1.68160808649729053898211840289, −0.57022588747924131615182560577,
0.973973985560210054859624720364, 2.04325776151440029301829325033, 2.57826000951774508594017075066, 3.97908113189217035425455021720, 4.33826924227061645563548630276, 5.28808586165357160690146539224, 5.74615949558397321273654608372, 6.71963448375343254550791464394, 7.47293875568047058349686186045, 8.018798052793064878721982525616
