| L(s) = 1 | + 5.29i·3-s − 11.1·5-s + 18.5i·7-s − 1.00·9-s + 11.8i·11-s − 40.2·13-s − 59.1i·15-s − 102.·17-s − 98.0·21-s + 125.·25-s + 137. i·27-s + 54·29-s − 62.6·33-s − 207. i·35-s − 212. i·39-s + ⋯ |
| L(s) = 1 | + 1.01i·3-s − 0.999·5-s + 0.999i·7-s − 0.0370·9-s + 0.324i·11-s − 0.858·13-s − 1.01i·15-s − 1.46·17-s − 1.01·21-s + 1.00·25-s + 0.980i·27-s + 0.345·29-s − 0.330·33-s − 0.999i·35-s − 0.874i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01381686027\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01381686027\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 11.1T \) |
| 7 | \( 1 - 18.5iT \) |
| good | 3 | \( 1 - 5.29iT - 27T^{2} \) |
| 11 | \( 1 - 11.8iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 40.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 102.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 - 54T + 2.43e4T^{2} \) |
| 31 | \( 1 + 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 - 6.89e4T^{2} \) |
| 43 | \( 1 + 7.95e4T^{2} \) |
| 47 | \( 1 + 619. iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 1.48e5T^{2} \) |
| 59 | \( 1 + 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 + 3.00e5T^{2} \) |
| 71 | \( 1 + 863. iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 523.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.38e3iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 47.6iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.65e3T + 9.12e5T^{2} \) |
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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
−10.97446742269934774313160447588, −10.14191678023177708248960077131, −9.205631740162166647846577570091, −8.611266321821291651449529602339, −7.49535237374166221032348866420, −6.53952725778026332426293031947, −5.08620032099521976642001963032, −4.54004713186749754670027663461, −3.48260769112308358706784146607, −2.23358587901637427027205855716,
0.00466066659332570932309269716, 1.09192184039771230722238159706, 2.57192885542672481855826070835, 3.97615223445963869253852007286, 4.76311010377699498829224649719, 6.42663675288941001808518713762, 7.09184252326596840360865359561, 7.71408930089632715994406201013, 8.520840712004308781724209507971, 9.726757750335555876684513357505
