L(s) = 1 | + (0.579 − 1.29i)2-s + (−1.32 − 1.49i)4-s + (0.667 + 0.667i)5-s + 7-s + (−2.69 + 0.848i)8-s + (1.24 − 0.474i)10-s + (1.57 − 1.57i)11-s + (−1.83 − 1.83i)13-s + (0.579 − 1.29i)14-s + (−0.468 + 3.97i)16-s − 3.40i·17-s + (3.18 − 3.18i)19-s + (0.110 − 1.88i)20-s + (−1.11 − 2.94i)22-s + 0.793i·23-s + ⋯ |
L(s) = 1 | + (0.409 − 0.912i)2-s + (−0.664 − 0.747i)4-s + (0.298 + 0.298i)5-s + 0.377·7-s + (−0.953 + 0.300i)8-s + (0.394 − 0.150i)10-s + (0.475 − 0.475i)11-s + (−0.507 − 0.507i)13-s + (0.154 − 0.344i)14-s + (−0.117 + 0.993i)16-s − 0.826i·17-s + (0.730 − 0.730i)19-s + (0.0247 − 0.421i)20-s + (−0.238 − 0.627i)22-s + 0.165i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.628 + 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.822028304\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.822028304\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.579 + 1.29i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 5 | \( 1 + (-0.667 - 0.667i)T + 5iT^{2} \) |
| 11 | \( 1 + (-1.57 + 1.57i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.83 + 1.83i)T + 13iT^{2} \) |
| 17 | \( 1 + 3.40iT - 17T^{2} \) |
| 19 | \( 1 + (-3.18 + 3.18i)T - 19iT^{2} \) |
| 23 | \( 1 - 0.793iT - 23T^{2} \) |
| 29 | \( 1 + (1.73 - 1.73i)T - 29iT^{2} \) |
| 31 | \( 1 + 3.28iT - 31T^{2} \) |
| 37 | \( 1 + (-7.72 + 7.72i)T - 37iT^{2} \) |
| 41 | \( 1 - 7.19T + 41T^{2} \) |
| 43 | \( 1 + (5.84 + 5.84i)T + 43iT^{2} \) |
| 47 | \( 1 + 13.0T + 47T^{2} \) |
| 53 | \( 1 + (-3.34 - 3.34i)T + 53iT^{2} \) |
| 59 | \( 1 + (7.41 - 7.41i)T - 59iT^{2} \) |
| 61 | \( 1 + (-1.93 - 1.93i)T + 61iT^{2} \) |
| 67 | \( 1 + (-6.38 + 6.38i)T - 67iT^{2} \) |
| 71 | \( 1 - 3.41iT - 71T^{2} \) |
| 73 | \( 1 - 8.13iT - 73T^{2} \) |
| 79 | \( 1 - 0.0502iT - 79T^{2} \) |
| 83 | \( 1 + (-2.29 - 2.29i)T + 83iT^{2} \) |
| 89 | \( 1 + 7.18T + 89T^{2} \) |
| 97 | \( 1 - 1.49T + 97T^{2} \) |
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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
−9.689490704277121050328170814465, −9.213799878864963651743755569834, −8.129047906978243685430476143971, −7.08055710122687432079415825726, −5.97520055423042371759141262756, −5.20564165951310510549291954817, −4.27181571543273614806295444683, −3.12345185472211368404415283888, −2.27327987932269027700805164093, −0.76996000219403853628788269836,
1.62918817722492504944058903608, 3.26110732324739930161157467472, 4.36139554961770709098457324200, 5.06137096860860797045838627350, 6.03680750463807458012628082971, 6.79568755993184465547031464883, 7.76030671252075791511585898531, 8.365595831503317520191997722033, 9.454894412705223457694080494768, 9.832143175168870659716762500258