| L(s) = 1 | + i·3-s − 4.20i·7-s − 9-s + 2.75·11-s − 0.753i·13-s + 4.96i·17-s − 4.75·19-s + 4.20·21-s + i·23-s − i·27-s − 4.96·29-s − 0.209·31-s + 2.75i·33-s + 5.71i·37-s + 0.753·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.59i·7-s − 0.333·9-s + 0.830·11-s − 0.208i·13-s + 1.20i·17-s − 1.09·19-s + 0.918·21-s + 0.208i·23-s − 0.192i·27-s − 0.921·29-s − 0.0375·31-s + 0.479i·33-s + 0.939i·37-s + 0.120·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9612491193\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9612491193\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 7 | \( 1 + 4.20iT - 7T^{2} \) |
| 11 | \( 1 - 2.75T + 11T^{2} \) |
| 13 | \( 1 + 0.753iT - 13T^{2} \) |
| 17 | \( 1 - 4.96iT - 17T^{2} \) |
| 19 | \( 1 + 4.75T + 19T^{2} \) |
| 29 | \( 1 + 4.96T + 29T^{2} \) |
| 31 | \( 1 + 0.209T + 31T^{2} \) |
| 37 | \( 1 - 5.71iT - 37T^{2} \) |
| 41 | \( 1 + 9.38T + 41T^{2} \) |
| 43 | \( 1 - 12.4iT - 43T^{2} \) |
| 47 | \( 1 + 7.17iT - 47T^{2} \) |
| 53 | \( 1 + 9.38iT - 53T^{2} \) |
| 59 | \( 1 - 4.96T + 59T^{2} \) |
| 61 | \( 1 + 5.17T + 61T^{2} \) |
| 67 | \( 1 - 0.209iT - 67T^{2} \) |
| 71 | \( 1 - 9.38T + 71T^{2} \) |
| 73 | \( 1 - 10.2iT - 73T^{2} \) |
| 79 | \( 1 - 2.41T + 79T^{2} \) |
| 83 | \( 1 - 5.45iT - 83T^{2} \) |
| 89 | \( 1 + 4.91T + 89T^{2} \) |
| 97 | \( 1 + 10.3iT - 97T^{2} \) |
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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
−8.251903836006784046596100430836, −7.51419595867275644352312911008, −6.62964977353206145423886253628, −6.32147405533709805506219002689, −5.20705283139938203746650109779, −4.45504547982695455139563027888, −3.79638087354649584403803918501, −3.44065134917038690036061163410, −1.99553792603682464508351726376, −1.07367719438634983631719053412,
0.24385565471411162343199355257, 1.71898196928071942447854102984, 2.29491963370830031195034529787, 3.12952953314253894517817089065, 4.11355415482380971274203267488, 5.06486958906800266946653421694, 5.69353424398966743663307948439, 6.34404053404787053762981466050, 6.96526564978894353759652716499, 7.69822169249867297592300995041
