L(s) = 1 | + 1.21i·5-s + (−0.355 − 2.62i)7-s − 4.06·11-s + 1.42·13-s + 4.74·17-s − 4.84·19-s + 6.03i·23-s + 3.51·25-s − 3.45·29-s + 3.15i·31-s + (3.19 − 0.432i)35-s − 9.24i·37-s + 1.24·41-s − 3.23i·43-s − 9.43·47-s + ⋯ |
L(s) = 1 | + 0.544i·5-s + (−0.134 − 0.990i)7-s − 1.22·11-s + 0.396·13-s + 1.15·17-s − 1.11·19-s + 1.25i·23-s + 0.703·25-s − 0.641·29-s + 0.566i·31-s + (0.539 − 0.0731i)35-s − 1.51i·37-s + 0.194·41-s − 0.492i·43-s − 1.37·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.469 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8047931434\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8047931434\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.355 + 2.62i)T \) |
good | 5 | \( 1 - 1.21iT - 5T^{2} \) |
| 11 | \( 1 + 4.06T + 11T^{2} \) |
| 13 | \( 1 - 1.42T + 13T^{2} \) |
| 17 | \( 1 - 4.74T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 - 6.03iT - 23T^{2} \) |
| 29 | \( 1 + 3.45T + 29T^{2} \) |
| 31 | \( 1 - 3.15iT - 31T^{2} \) |
| 37 | \( 1 + 9.24iT - 37T^{2} \) |
| 41 | \( 1 - 1.24T + 41T^{2} \) |
| 43 | \( 1 + 3.23iT - 43T^{2} \) |
| 47 | \( 1 + 9.43T + 47T^{2} \) |
| 53 | \( 1 - 2.44T + 53T^{2} \) |
| 59 | \( 1 - 10.2iT - 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 6.51iT - 67T^{2} \) |
| 71 | \( 1 + 2.14iT - 71T^{2} \) |
| 73 | \( 1 - 12.6iT - 73T^{2} \) |
| 79 | \( 1 - 10.6T + 79T^{2} \) |
| 83 | \( 1 - 16.3iT - 83T^{2} \) |
| 89 | \( 1 - 14.9T + 89T^{2} \) |
| 97 | \( 1 - 11.4iT - 97T^{2} \) |
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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
−8.578986476565727962327570457201, −7.73285278310924631383684557300, −7.37316521051478518883674185849, −6.56764030269718031604827325259, −5.69686637185045279462877612020, −5.02416484947140722174524832458, −3.92700928743899363483325801850, −3.35646994635338888743755528808, −2.38721647512335798402863191576, −1.16365212826125576618515872607,
0.23928451312791489292721629604, 1.69163903573784003930260582168, 2.65831877663432313392325844435, 3.40119625050673735119740341585, 4.71909400600835632284172309819, 5.06245026371734678993608110232, 6.05594080035129470894038772908, 6.48462197057830347117884835948, 7.80006591779177219903467279508, 8.163428856057764225469448109833