L(s) = 1 | − i·2-s + 0.618i·3-s − 4-s + 0.618·6-s + 1.61i·7-s + i·8-s + 0.618·9-s − 0.618i·12-s + 1.61·14-s + 16-s − 0.618i·18-s − 1.00·21-s − 1.61i·23-s − 0.618·24-s + i·27-s − 1.61i·28-s + ⋯ |
L(s) = 1 | − i·2-s + 0.618i·3-s − 4-s + 0.618·6-s + 1.61i·7-s + i·8-s + 0.618·9-s − 0.618i·12-s + 1.61·14-s + 16-s − 0.618i·18-s − 1.00·21-s − 1.61i·23-s − 0.618·24-s + i·27-s − 1.61i·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8220717437\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8220717437\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.618iT - T^{2} \) |
| 7 | \( 1 - 1.61iT - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.61iT - T^{2} \) |
| 29 | \( 1 + 0.618T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - 0.618T + T^{2} \) |
| 43 | \( 1 - 0.618iT - T^{2} \) |
| 47 | \( 1 + 0.618iT - T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + 1.61T + T^{2} \) |
| 67 | \( 1 + 2iT - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.61iT - T^{2} \) |
| 89 | \( 1 - 1.61T + T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−11.05528460054153254276676768603, −10.34004485497887229454835413743, −9.367335642240957382684678719080, −8.930074537195376848494709426890, −7.903753242028869966867672037200, −6.26445790760845398235712485742, −5.18998964462435297827539165020, −4.37510281737693269411401181871, −3.10704401620495720348084907954, −2.00692432004278416078375153215,
1.25276422819773355308339692320, 3.66735397126259653442367922964, 4.48356678306683325587443462193, 5.75840755322082570063539996459, 6.88949502527873404401751038917, 7.37778346034146958242057896319, 8.006120872951264265066786573584, 9.348832613678366874757574880527, 10.07346951200950418173798424667, 11.00932626690238060210699390275