| L(s) = 1 | + 1.73·3-s + (−1.77 − 4.67i)5-s − 2.37·7-s + 2.99·9-s + 1.43i·11-s − 13.7i·13-s + (−3.07 − 8.09i)15-s − 2.95i·17-s + 5.04i·19-s − 4.10·21-s − 23.2·23-s + (−18.7 + 16.5i)25-s + 5.19·27-s + 9.07·29-s − 6.79i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + (−0.354 − 0.935i)5-s − 0.338·7-s + 0.333·9-s + 0.130i·11-s − 1.05i·13-s + (−0.204 − 0.539i)15-s − 0.174i·17-s + 0.265i·19-s − 0.195·21-s − 1.01·23-s + (−0.748 + 0.663i)25-s + 0.192·27-s + 0.312·29-s − 0.219i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.935 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9471170787\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9471170787\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 + (1.77 + 4.67i)T \) |
| good | 7 | \( 1 + 2.37T + 49T^{2} \) |
| 11 | \( 1 - 1.43iT - 121T^{2} \) |
| 13 | \( 1 + 13.7iT - 169T^{2} \) |
| 17 | \( 1 + 2.95iT - 289T^{2} \) |
| 19 | \( 1 - 5.04iT - 361T^{2} \) |
| 23 | \( 1 + 23.2T + 529T^{2} \) |
| 29 | \( 1 - 9.07T + 841T^{2} \) |
| 31 | \( 1 + 6.79iT - 961T^{2} \) |
| 37 | \( 1 + 32.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 44.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 11.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 74.9T + 2.20e3T^{2} \) |
| 53 | \( 1 + 29.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 34.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 100.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 95.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 130. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 47.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 87.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 24.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + 145.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 135. iT - 9.40e3T^{2} \) |
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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
−9.549361764343592524947481324715, −8.392846276257188342360184913212, −8.096608458471042109393955939791, −7.08320858184757988661411825443, −5.91490212894451862239140426531, −5.00266773835462673727998657232, −3.99881614723186627255131170492, −3.08498407326173032273221233302, −1.70572609466325159189140109922, −0.26475626369777710835039904990,
1.80094200762783155436404169083, 2.94117377012490774453402283123, 3.75844883917786849311187539200, 4.74406020879466896251389186336, 6.26533519327480280057274789264, 6.74779601954370747136724427189, 7.73098963201829767652204965920, 8.451772397429340906507899659451, 9.458811688448894306795504984713, 10.09410092922527386974191499600
