L(s) = 1 | + 1.41i·3-s + 1.41i·5-s − 1.41i·7-s − 1.00·9-s − 11-s + 13-s − 2.00·15-s − 17-s + 2.00·21-s + 23-s − 1.00·25-s − 1.41i·29-s + 31-s − 1.41i·33-s + 2.00·35-s + ⋯ |
L(s) = 1 | + 1.41i·3-s + 1.41i·5-s − 1.41i·7-s − 1.00·9-s − 11-s + 13-s − 2.00·15-s − 17-s + 2.00·21-s + 23-s − 1.00·25-s − 1.41i·29-s + 31-s − 1.41i·33-s + 2.00·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7856528013\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7856528013\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 - 1.41iT - T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 + 1.41iT - T^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 - T + T^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 - T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + 1.41iT - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T + T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + T + T^{2} \) |
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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
−11.24865578968259949896356557230, −10.90191967278806697722992954417, −10.32833321386572454806022913246, −9.591838385765658656122194205426, −8.230301142341659719610944219166, −7.15850105205965188666751717656, −6.22936222392466156268934185526, −4.72880168590109189823569378385, −3.85198647322505393678844714187, −2.88505902030089217973245696291,
1.42520825299580565134621568968, 2.69832019129118238778427440776, 4.81266998115598006152831708944, 5.68473255032889856435653275854, 6.64564911634403557367843557597, 7.947597509374560353552093530580, 8.632041500138724357318344480768, 9.160599214552369478581250932865, 10.85300882763172527173680091123, 11.89811014839511366047574208218