| L(s) = 1 | + 0.162·5-s + i·7-s − 2.40i·11-s + 4.76i·13-s − 2.74i·17-s + 1.86·19-s + 1.32·23-s − 4.97·25-s + 3.96·29-s − 8.01i·31-s + 0.162i·35-s + 6.09i·37-s − 3.16i·41-s + 1.70·43-s + 12.1·47-s + ⋯ |
| L(s) = 1 | + 0.0724·5-s + 0.377i·7-s − 0.724i·11-s + 1.32i·13-s − 0.665i·17-s + 0.426·19-s + 0.277·23-s − 0.994·25-s + 0.735·29-s − 1.43i·31-s + 0.0273i·35-s + 1.00i·37-s − 0.494i·41-s + 0.260·43-s + 1.77·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.108i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.936308224\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.936308224\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 0.162T + 5T^{2} \) |
| 11 | \( 1 + 2.40iT - 11T^{2} \) |
| 13 | \( 1 - 4.76iT - 13T^{2} \) |
| 17 | \( 1 + 2.74iT - 17T^{2} \) |
| 19 | \( 1 - 1.86T + 19T^{2} \) |
| 23 | \( 1 - 1.32T + 23T^{2} \) |
| 29 | \( 1 - 3.96T + 29T^{2} \) |
| 31 | \( 1 + 8.01iT - 31T^{2} \) |
| 37 | \( 1 - 6.09iT - 37T^{2} \) |
| 41 | \( 1 + 3.16iT - 41T^{2} \) |
| 43 | \( 1 - 1.70T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 8.22T + 53T^{2} \) |
| 59 | \( 1 + 5.45iT - 59T^{2} \) |
| 61 | \( 1 - 4.28iT - 61T^{2} \) |
| 67 | \( 1 - 4.61T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 6.37T + 73T^{2} \) |
| 79 | \( 1 + 9.75iT - 79T^{2} \) |
| 83 | \( 1 - 16.4iT - 83T^{2} \) |
| 89 | \( 1 - 2.35iT - 89T^{2} \) |
| 97 | \( 1 - 14.4T + 97T^{2} \) |
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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
−8.083175005890027409236675747456, −7.40392138293381456617817528484, −6.59343940776711009177062674189, −6.01218828740992335167534807125, −5.25533104305342782130927692921, −4.43870262470863864044139722785, −3.68779229437927462036120586692, −2.72620380515324566840891011697, −1.95222047008420591932174812245, −0.73664487090098946585606027851,
0.73420279294321303951971219508, 1.79010178076512566671565412106, 2.82627235015425540350587982762, 3.60578433236467347657302025903, 4.43055025739323112984750281139, 5.23348056547866613640805161530, 5.87349473945451243622011503461, 6.69530130791490349966509592280, 7.46912172802507451303463318559, 7.923304634790753890165650994295
