L(s) = 1 | + i·2-s + 4-s − 4i·7-s + 3i·8-s − 11-s − 2i·13-s + 4·14-s − 16-s − 2i·17-s − i·22-s − 8i·23-s + 2·26-s − 4i·28-s − 6·29-s − 8·31-s + 5i·32-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.5·4-s − 1.51i·7-s + 1.06i·8-s − 0.301·11-s − 0.554i·13-s + 1.06·14-s − 0.250·16-s − 0.485i·17-s − 0.213i·22-s − 1.66i·23-s + 0.392·26-s − 0.755i·28-s − 1.11·29-s − 1.43·31-s + 0.883i·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.543697105\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.543697105\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - iT - 2T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 8iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 14iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.572118508735431150178297917237, −7.65425551010714806031984579471, −7.39237206538559556034207148451, −6.62739717935609349818089283314, −5.81035351536980736913727158654, −4.96266235526207981575931562842, −4.05335386818013800255305889111, −3.05111798410349557260718715854, −1.93158060445720587440926283626, −0.46773758599670833236951919248,
1.59218125761055456862938865920, 2.22233493526386804438013593533, 3.18560476081113685080029332595, 3.96787474701537084908378451881, 5.35436801606081535945364348358, 5.79021730882807693922797072594, 6.76075008149238254837919490492, 7.52416623741558427761001439276, 8.426414204575432439515965727940, 9.313090476492710248391318116837