L(s) = 1 | + (−0.5 − 0.866i)3-s + (1 − 1.73i)5-s + (−0.499 + 0.866i)9-s + 2·13-s − 1.99·15-s + (3 + 5.19i)17-s + (2 − 3.46i)19-s + (−2 + 3.46i)23-s + (0.500 + 0.866i)25-s + 0.999·27-s + 6·29-s + (4 + 6.92i)31-s + (5 − 8.66i)37-s + (−1 − 1.73i)39-s + 10·41-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.447 − 0.774i)5-s + (−0.166 + 0.288i)9-s + 0.554·13-s − 0.516·15-s + (0.727 + 1.26i)17-s + (0.458 − 0.794i)19-s + (−0.417 + 0.722i)23-s + (0.100 + 0.173i)25-s + 0.192·27-s + 1.11·29-s + (0.718 + 1.24i)31-s + (0.821 − 1.42i)37-s + (−0.160 − 0.277i)39-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.900148475\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.900148475\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-3 - 5.19i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 + 12T + 43T^{2} \) |
| 47 | \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 - 10.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.753914993918196851662717852248, −8.226761406041811137019918247963, −7.34711857467770339355663164725, −6.47418097622405755482213710068, −5.72726052657052670948970775232, −5.12136295921012204264526645214, −4.10400451830770355599843174279, −3.02151309939763951578538775208, −1.73191280267095599250380552458, −0.929714966932420943581078651768,
0.977066677685226687078752513273, 2.54790336892662606531899640750, 3.21231994810979406353241313101, 4.32873713137061022463200510505, 5.09969430861507248203205321650, 6.25513408540852829841041230575, 6.35109953824393146174762762268, 7.63926877739440878427576325625, 8.209360898215080996502146761162, 9.362245400259351899539144120333