| L(s) = 1 | + (−0.730 − 1.57i)3-s − 1.25·7-s + (−1.93 + 2.29i)9-s − 3.02i·11-s + 5.65·13-s − 2.45·17-s + 1.77·19-s + (0.916 + 1.97i)21-s − 8.84i·23-s + (5.01 + 1.36i)27-s − 3.79·29-s + 5.19i·31-s + (−4.74 + 2.20i)33-s + 6.45·37-s + (−4.12 − 8.88i)39-s + ⋯ |
| L(s) = 1 | + (−0.421 − 0.906i)3-s − 0.474·7-s + (−0.644 + 0.764i)9-s − 0.911i·11-s + 1.56·13-s − 0.595·17-s + 0.407·19-s + (0.200 + 0.430i)21-s − 1.84i·23-s + (0.964 + 0.262i)27-s − 0.704·29-s + 0.933i·31-s + (−0.826 + 0.384i)33-s + 1.06·37-s + (−0.661 − 1.42i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9661468803\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9661468803\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.730 + 1.57i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 1.25T + 7T^{2} \) |
| 11 | \( 1 + 3.02iT - 11T^{2} \) |
| 13 | \( 1 - 5.65T + 13T^{2} \) |
| 17 | \( 1 + 2.45T + 17T^{2} \) |
| 19 | \( 1 - 1.77T + 19T^{2} \) |
| 23 | \( 1 + 8.84iT - 23T^{2} \) |
| 29 | \( 1 + 3.79T + 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 6.45T + 37T^{2} \) |
| 41 | \( 1 - 7.57iT - 41T^{2} \) |
| 43 | \( 1 + 4.37iT - 43T^{2} \) |
| 47 | \( 1 + 1.83iT - 47T^{2} \) |
| 53 | \( 1 + 12.0iT - 53T^{2} \) |
| 59 | \( 1 - 4.91iT - 59T^{2} \) |
| 61 | \( 1 + 8.16iT - 61T^{2} \) |
| 67 | \( 1 + 8.50iT - 67T^{2} \) |
| 71 | \( 1 + 7.00T + 71T^{2} \) |
| 73 | \( 1 + 4.59iT - 73T^{2} \) |
| 79 | \( 1 + 7.36iT - 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + 3.65iT - 89T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−8.474831414426653970747112834224, −8.006267991638317636461316067542, −6.81638341690239565074932759955, −6.38130251599327547146249686883, −5.77810145416903870395685820009, −4.75739182294000339984876313400, −3.58824250152806256396588615543, −2.72661798896971666230361110934, −1.47705423775977237433267042411, −0.37123129607618338165161054355,
1.35533624130176671063556107727, 2.84224986906922580852491488014, 3.83959284104059735450382636970, 4.31404348874679200758881313946, 5.53565902625899023168703097025, 5.94677816234943026888278223670, 6.90062015805592808133597405918, 7.75330636363171622777352543126, 8.772809140290917738066645338021, 9.456107014763596454139381393204
