L(s) = 1 | − 1.41i·2-s + (−0.5 + 2.59i)7-s − 2.82i·8-s − 1.41i·11-s + 5.19i·13-s + (3.67 + 0.707i)14-s − 4.00·16-s + 7.34·17-s − 5.19i·19-s − 2.00·22-s + 7.07i·23-s + 7.34·26-s − 1.41i·29-s + 5.19i·31-s − 10.3i·34-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + (−0.188 + 0.981i)7-s − 0.999i·8-s − 0.426i·11-s + 1.44i·13-s + (0.981 + 0.188i)14-s − 1.00·16-s + 1.78·17-s − 1.19i·19-s − 0.426·22-s + 1.47i·23-s + 1.44·26-s − 0.262i·29-s + 0.933i·31-s − 1.78i·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.965492438\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.965492438\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.5 - 2.59i)T \) |
good | 2 | \( 1 + 1.41iT - 2T^{2} \) |
| 11 | \( 1 + 1.41iT - 11T^{2} \) |
| 13 | \( 1 - 5.19iT - 13T^{2} \) |
| 17 | \( 1 - 7.34T + 17T^{2} \) |
| 19 | \( 1 + 5.19iT - 19T^{2} \) |
| 23 | \( 1 - 7.07iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 - 8T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 - 7.34T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 7.34T + 59T^{2} \) |
| 61 | \( 1 + 5.19iT - 61T^{2} \) |
| 67 | \( 1 + 7T + 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 10.3iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 7.34T + 83T^{2} \) |
| 89 | \( 1 - 14.6T + 89T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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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
−9.428637791516586588657662983583, −8.912405007858579811000494076777, −7.73319924978320057180608151471, −6.88939710659647541474868346019, −6.02974905111421388351410693684, −5.15916932600175843795756897853, −3.93694846463285587294594276301, −3.09910890846450230163799822959, −2.24737770046375000083303385871, −1.13630848281710471679405501188,
0.940149249540689580617938651699, 2.56793036557892588628104398178, 3.65008215603560782964561010883, 4.71816893428223097363821351136, 5.75473952560769904379158503179, 6.17632868371731035056422726068, 7.37883807629813188162529688174, 7.70116694471963126751880835765, 8.299939476372337503160887164242, 9.541808900039462334915150196969