| L(s) = 1 | + (1.12 + 1.31i)3-s − 4.34·7-s + (−0.448 + 2.96i)9-s − 1.83i·11-s + 0.588·13-s + 5.37·17-s − 5.38·19-s + (−4.90 − 5.70i)21-s − 2.40i·23-s + (−4.40 + 2.76i)27-s − 7.98·29-s − 7.06i·31-s + (2.41 − 2.07i)33-s − 2.72·37-s + (0.664 + 0.772i)39-s + ⋯ |
| L(s) = 1 | + (0.652 + 0.758i)3-s − 1.64·7-s + (−0.149 + 0.988i)9-s − 0.553i·11-s + 0.163·13-s + 1.30·17-s − 1.23·19-s + (−1.07 − 1.24i)21-s − 0.502i·23-s + (−0.847 + 0.531i)27-s − 1.48·29-s − 1.26i·31-s + (0.419 − 0.361i)33-s − 0.447·37-s + (0.106 + 0.123i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5491684576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5491684576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.12 - 1.31i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.34T + 7T^{2} \) |
| 11 | \( 1 + 1.83iT - 11T^{2} \) |
| 13 | \( 1 - 0.588T + 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 + 5.38T + 19T^{2} \) |
| 23 | \( 1 + 2.40iT - 23T^{2} \) |
| 29 | \( 1 + 7.98T + 29T^{2} \) |
| 31 | \( 1 + 7.06iT - 31T^{2} \) |
| 37 | \( 1 + 2.72T + 37T^{2} \) |
| 41 | \( 1 + 3.42iT - 41T^{2} \) |
| 43 | \( 1 - 2.96iT - 43T^{2} \) |
| 47 | \( 1 - 9.81iT - 47T^{2} \) |
| 53 | \( 1 + 6.65iT - 53T^{2} \) |
| 59 | \( 1 + 10.7iT - 59T^{2} \) |
| 61 | \( 1 + 9.27iT - 61T^{2} \) |
| 67 | \( 1 + 4.13iT - 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 + 4.42iT - 73T^{2} \) |
| 79 | \( 1 + 12.5iT - 79T^{2} \) |
| 83 | \( 1 + 11.5T + 83T^{2} \) |
| 89 | \( 1 + 4.21iT - 89T^{2} \) |
| 97 | \( 1 - 2.16iT - 97T^{2} \) |
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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
−8.906214192301878138365599237686, −8.091986097607130656772854693781, −7.32613940718080227430773104229, −6.22221031147748224199991382131, −5.77100684673181477410472406477, −4.57432096557242892694733790112, −3.58835232534638193909129860712, −3.22652568017563742207797494344, −2.11095496093722636300723166083, −0.16376714431215159176568783484,
1.37652200769335939012314211132, 2.54889319238154786964487565658, 3.37717452441869346802279278664, 4.01999639719396010562879124425, 5.55085618012248038433106782956, 6.17828937478332758105426187581, 7.09403537726034482123609034425, 7.37125342687651089826674318949, 8.569061788029305269018054164956, 9.024176502311855834064537682460
