L(s) = 1 | − 1.41i·2-s − 2.00·4-s − 2·7-s + 2.82i·8-s − 5.65i·11-s + 2.82i·14-s + 4.00·16-s − 8.00·22-s + 4.00·28-s + 2.82i·29-s − 10·31-s − 5.65i·32-s + 11.3i·44-s − 3·49-s + 14.1i·53-s + ⋯ |
L(s) = 1 | − 0.999i·2-s − 1.00·4-s − 0.755·7-s + 1.00i·8-s − 1.70i·11-s + 0.755i·14-s + 1.00·16-s − 1.70·22-s + 0.755·28-s + 0.525i·29-s − 1.79·31-s − 1.00i·32-s + 1.70i·44-s − 0.428·49-s + 1.94i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 1.41iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 10T + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14.1iT - 53T^{2} \) |
| 59 | \( 1 - 11.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 10T + 79T^{2} \) |
| 83 | \( 1 - 5.65iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.947174569086627530166685160944, −8.186389872541100225267746582602, −7.16980034059035337689784753713, −5.97841109450943491439673499190, −5.48695190598253148257586371512, −4.22532595064622352166774962568, −3.38481122187681106371870946495, −2.76014854368415329163206817809, −1.30825775781324601032672761017, 0,
1.87994880808488266813241989632, 3.36275080870364268618296400089, 4.27738163237587234390225253332, 5.08098196072103224937510906616, 5.95937510888850869399184321810, 6.87201030730388695291060964067, 7.27555897520815826322214297244, 8.159503348502677388655341050829, 9.097278333646016043644896632751