| L(s) = 1 | + (1.45 − 1.67i)2-s + (0.841 − 0.540i)3-s + (−0.415 − 2.88i)4-s + (−0.415 − 0.909i)5-s + (0.315 − 2.19i)6-s + (−1.14 + 0.337i)7-s + (−1.71 − 1.10i)8-s + (0.415 − 0.909i)9-s + (−2.12 − 0.624i)10-s + (−1.24 − 1.44i)11-s + (−1.91 − 2.20i)12-s + (1.06 + 0.313i)13-s + (−1.10 + 2.41i)14-s + (−0.841 − 0.540i)15-s + (1.26 − 0.371i)16-s + (0.404 − 2.81i)17-s + ⋯ |
| L(s) = 1 | + (1.02 − 1.18i)2-s + (0.485 − 0.312i)3-s + (−0.207 − 1.44i)4-s + (−0.185 − 0.406i)5-s + (0.128 − 0.896i)6-s + (−0.433 + 0.127i)7-s + (−0.606 − 0.389i)8-s + (0.138 − 0.303i)9-s + (−0.672 − 0.197i)10-s + (−0.376 − 0.434i)11-s + (−0.551 − 0.636i)12-s + (0.296 + 0.0870i)13-s + (−0.294 + 0.645i)14-s + (−0.217 − 0.139i)15-s + (0.316 − 0.0928i)16-s + (0.0980 − 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 345 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.562 + 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12601 - 2.12933i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12601 - 2.12933i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.841 + 0.540i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 23 | \( 1 + (-1.92 - 4.39i)T \) |
| good | 2 | \( 1 + (-1.45 + 1.67i)T + (-0.284 - 1.97i)T^{2} \) |
| 7 | \( 1 + (1.14 - 0.337i)T + (5.88 - 3.78i)T^{2} \) |
| 11 | \( 1 + (1.24 + 1.44i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 0.313i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.404 + 2.81i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.743 - 5.17i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (0.516 - 3.59i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.80 - 2.44i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (1.35 - 2.96i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (2.05 + 4.50i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (3.15 - 2.02i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 9.16T + 47T^{2} \) |
| 53 | \( 1 + (-3.35 + 0.986i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (4.67 + 1.37i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (6.76 + 4.34i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (3.91 - 4.51i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (-3.58 + 4.14i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-0.943 - 6.56i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (9.78 + 2.87i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-2.44 + 5.34i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (7.19 - 4.62i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-0.672 - 1.47i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−11.49531074876645557958575845245, −10.47682239574142531888820742641, −9.580585146133058070071640417616, −8.508795053475759713393590340539, −7.41868083229911707614645944144, −5.95187264899702159502352950727, −4.96453332566078759370561137411, −3.68622403585629592073496448790, −2.92164165484059375236625366850, −1.42984237999388628656039396488,
2.80366002106155087461568091578, 3.96837296104528912199419920403, 4.85379543800041819711954720380, 6.07541086200648707029975533697, 6.93166911998752986464158205574, 7.76845040235685597768615727857, 8.731088683723937900842679216779, 9.978745149133017355566285892337, 10.85173362863935083997427898828, 12.18649515147172235777542339992
