L(s) = 1 | + (−1 − i)2-s − 3-s + 2i·4-s + (1 + i)6-s − 2i·7-s + (2 − 2i)8-s + 9-s + 4i·11-s − 2i·12-s + (−2 + 2i)14-s − 4·16-s + 6i·17-s + (−1 − i)18-s − 4i·19-s + 2i·21-s + (4 − 4i)22-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s − 0.577·3-s + i·4-s + (0.408 + 0.408i)6-s − 0.755i·7-s + (0.707 − 0.707i)8-s + 0.333·9-s + 1.20i·11-s − 0.577i·12-s + (−0.534 + 0.534i)14-s − 16-s + 1.45i·17-s + (−0.235 − 0.235i)18-s − 0.917i·19-s + 0.436i·21-s + (0.852 − 0.852i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.807553 - 0.131047i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.807553 - 0.131047i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 2iT - 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 4iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 4iT - 47T^{2} \) |
| 53 | \( 1 - 10T + 53T^{2} \) |
| 59 | \( 1 + 8iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 4T + 71T^{2} \) |
| 73 | \( 1 - 10iT - 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−10.57884674361779277192378275215, −10.00926917532685268603592528672, −9.055172867428581192267337003432, −8.012297790898368632558712994195, −7.16415598959468653618847683069, −6.36746815172280571191372075579, −4.69355085039934156189379363896, −4.02877283626781030088204718150, −2.46748202581591711568639222369, −1.04615743955109003977003270597,
0.830475810933722381581175432488, 2.63746423885872204149972438473, 4.44193463294843333647798209742, 5.76845275838164810230677402759, 5.90531277092300255993086856304, 7.21049533137671014065508425666, 8.051205548668110708012665065461, 8.952777053694957982353977851913, 9.704584113283425587253663132179, 10.57861898893650757886255015839