| L(s) = 1 | − 7.65i·5-s − 1.65i·7-s − 1.51·11-s + 0.343i·13-s + 13.3·17-s + 20.8·19-s − 33.6i·23-s − 33.6·25-s + 39.6i·29-s − 45.2i·31-s − 12.6·35-s − 29.5i·37-s + 24.6·41-s + 50.0·43-s − 35.3i·47-s + ⋯ |
| L(s) = 1 | − 1.53i·5-s − 0.236i·7-s − 0.137·11-s + 0.0263i·13-s + 0.783·17-s + 1.09·19-s − 1.46i·23-s − 1.34·25-s + 1.36i·29-s − 1.45i·31-s − 0.362·35-s − 0.799i·37-s + 0.600·41-s + 1.16·43-s − 0.751i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.855449127\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855449127\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 7.65iT - 25T^{2} \) |
| 7 | \( 1 + 1.65iT - 49T^{2} \) |
| 11 | \( 1 + 1.51T + 121T^{2} \) |
| 13 | \( 1 - 0.343iT - 169T^{2} \) |
| 17 | \( 1 - 13.3T + 289T^{2} \) |
| 19 | \( 1 - 20.8T + 361T^{2} \) |
| 23 | \( 1 + 33.6iT - 529T^{2} \) |
| 29 | \( 1 - 39.6iT - 841T^{2} \) |
| 31 | \( 1 + 45.2iT - 961T^{2} \) |
| 37 | \( 1 + 29.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 24.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 50.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 35.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 16.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 53.1T + 3.48e3T^{2} \) |
| 61 | \( 1 + 34.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 62.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 40.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 55.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 137. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 114.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 2.56T + 7.92e3T^{2} \) |
| 97 | \( 1 + 138.T + 9.40e3T^{2} \) |
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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
−8.673967945413040904522063103674, −7.79743460286860523718498358419, −7.20218168732604232265214300734, −5.94056229119152633150304105712, −5.35598174836839907981909303041, −4.55837929306307406001396538686, −3.81024389720489197490987398561, −2.55996539578879918508988376091, −1.27459521408495924607353258048, −0.50410706476296196122522386337,
1.29074940513651716218377819777, 2.64330274121428607418480361820, 3.17415157186922409851518203271, 4.10022501872830995270130090367, 5.43198461365032634260418817809, 5.93052562539039821916362470584, 6.96056072745967570463687273727, 7.44190450662099803549934375238, 8.151004669654584377680068912686, 9.303355263802337096610840717502
