L(s) = 1 | + 0.618i·3-s − i·7-s + 0.618·9-s − i·11-s + 0.618·13-s − 17-s − 0.618i·19-s + 0.618·21-s + i·27-s − 29-s − 1.61i·31-s + 0.618·33-s + 0.381i·39-s + 41-s − i·43-s + ⋯ |
L(s) = 1 | + 0.618i·3-s − i·7-s + 0.618·9-s − i·11-s + 0.618·13-s − 17-s − 0.618i·19-s + 0.618·21-s + i·27-s − 29-s − 1.61i·31-s + 0.618·33-s + 0.381i·39-s + 41-s − i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.279793340\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.279793340\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 - 0.618iT - T^{2} \) |
| 7 | \( 1 + iT - T^{2} \) |
| 11 | \( 1 + iT - T^{2} \) |
| 13 | \( 1 - 0.618T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + 0.618iT - T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + T + T^{2} \) |
| 31 | \( 1 + 1.61iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 43 | \( 1 + iT - T^{2} \) |
| 47 | \( 1 + 1.61iT - T^{2} \) |
| 53 | \( 1 + 1.61T + T^{2} \) |
| 59 | \( 1 - 1.61iT - T^{2} \) |
| 61 | \( 1 - 0.618T + T^{2} \) |
| 67 | \( 1 - 1.61iT - T^{2} \) |
| 71 | \( 1 + iT - T^{2} \) |
| 73 | \( 1 + 1.61T + T^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - 1.61T + T^{2} \) |
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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
−8.798808739723886242674689971012, −7.67002450574604537419499466069, −7.18401611918434118011068748555, −6.30221258020299763965376251192, −5.57281628010657758576924499425, −4.51178636250351241981734582257, −4.02583803413308960917667996309, −3.33921066949657398906188169899, −2.06816803820930480604423905075, −0.74316149484627271561463929145,
1.49857196349192244242743947173, 2.10026071645969598454327607370, 3.16281552439781541146264152255, 4.27921128706487759394645745623, 4.92229870832455759520692965289, 5.97569036803006198942424979041, 6.48323206999738062263551776983, 7.27773933945870132432609727737, 7.911185273167649129787707467997, 8.697862580920212417814460473286