| L(s) = 1 | + i·2-s + (−1.06 − 1.36i)3-s − 4-s + (1.36 − 1.06i)6-s + (−2.62 + 0.294i)7-s − i·8-s + (−0.716 + 2.91i)9-s + 1.43i·11-s + (1.06 + 1.36i)12-s − 4.73i·13-s + (−0.294 − 2.62i)14-s + 16-s + 2.59·17-s + (−2.91 − 0.716i)18-s + 2.72i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.616 − 0.786i)3-s − 0.5·4-s + (0.556 − 0.436i)6-s + (−0.993 + 0.111i)7-s − 0.353i·8-s + (−0.238 + 0.971i)9-s + 0.431i·11-s + (0.308 + 0.393i)12-s − 1.31i·13-s + (−0.0786 − 0.702i)14-s + 0.250·16-s + 0.630·17-s + (−0.686 − 0.168i)18-s + 0.625i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9437714898\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9437714898\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (1.06 + 1.36i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.62 - 0.294i)T \) |
| good | 11 | \( 1 - 1.43iT - 11T^{2} \) |
| 13 | \( 1 + 4.73iT - 13T^{2} \) |
| 17 | \( 1 - 2.59T + 17T^{2} \) |
| 19 | \( 1 - 2.72iT - 19T^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + 2.82iT - 29T^{2} \) |
| 31 | \( 1 - 5.91iT - 31T^{2} \) |
| 37 | \( 1 + 2.39T + 37T^{2} \) |
| 41 | \( 1 - 11.1T + 41T^{2} \) |
| 43 | \( 1 - 0.432T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 5.69iT - 53T^{2} \) |
| 59 | \( 1 - 10.7T + 59T^{2} \) |
| 61 | \( 1 - 1.05iT - 61T^{2} \) |
| 67 | \( 1 - 9.82T + 67T^{2} \) |
| 71 | \( 1 - 13.2iT - 71T^{2} \) |
| 73 | \( 1 + 7.58iT - 73T^{2} \) |
| 79 | \( 1 - 16.5T + 79T^{2} \) |
| 83 | \( 1 + 12.9T + 83T^{2} \) |
| 89 | \( 1 - 3.90T + 89T^{2} \) |
| 97 | \( 1 + 14.0iT - 97T^{2} \) |
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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
−10.07700107822890469814545501830, −9.098524784431458728006485488752, −8.020855340682110613426746068840, −7.46161890817946616027318618821, −6.64301195367793424125345519106, −5.76331471531360335735047393909, −5.32863424549491771023962356059, −3.87038434673520163459272571979, −2.64859415557668962745311956701, −0.899100208035858701000019826264,
0.65543857291516831369126118531, 2.53454687898257615647826247983, 3.66493145016444511028379216013, 4.30049671674795271480570646785, 5.41569933059396087679695927689, 6.27713126909328764531548821034, 7.11520764337747543372180909857, 8.561010257841238859231371143067, 9.377738072998315439750408324351, 9.740169262091020519249875401008
