L(s) = 1 | − i·3-s + (−2 + i)5-s + 2i·7-s − 9-s + 6·11-s − 2i·13-s + (1 + 2i)15-s + 6i·17-s + 4·19-s + 2·21-s + 8i·23-s + (3 − 4i)25-s + i·27-s + 8·31-s − 6i·33-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.894 + 0.447i)5-s + 0.755i·7-s − 0.333·9-s + 1.80·11-s − 0.554i·13-s + (0.258 + 0.516i)15-s + 1.45i·17-s + 0.917·19-s + 0.436·21-s + 1.66i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s + 1.43·31-s − 1.04i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24964 + 0.295000i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24964 + 0.295000i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2 - i)T \) |
good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 2iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 + 6T + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 12iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 14T + 89T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−11.41001615382701068271221791457, −10.21254405922517089727111507048, −9.089200700241600988414189820745, −8.311413955675650882915981570292, −7.40762907114216634268869508007, −6.49432313314898088550576780150, −5.63002845796353315854045755264, −4.04637574187906573675553715947, −3.13915210839095256579411462251, −1.44886843375653225317814170473,
0.939682620990262182447286749048, 3.18964128047816335396606125578, 4.27245468657859865509088629692, 4.77497765649206650674436626629, 6.46485624881035546648702512432, 7.20885547727058697242464588969, 8.362349536518815265263694115843, 9.192556465894648816523450543746, 9.899983890173253831612884825828, 11.10193933423440666975330015196