| L(s) = 1 | − i·3-s − 0.585i·5-s + 3.41·7-s − 9-s + 2i·11-s + 2.82i·13-s − 0.585·15-s + 3.65·17-s + 5.65i·19-s − 3.41i·21-s + 1.17·23-s + 4.65·25-s + i·27-s + 0.585i·29-s − 4.58·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.261i·5-s + 1.29·7-s − 0.333·9-s + 0.603i·11-s + 0.784i·13-s − 0.151·15-s + 0.886·17-s + 1.29i·19-s − 0.745i·21-s + 0.244·23-s + 0.931·25-s + 0.192i·27-s + 0.108i·29-s − 0.823·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.998721655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.998721655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 + 0.585iT - 5T^{2} \) |
| 7 | \( 1 - 3.41T + 7T^{2} \) |
| 11 | \( 1 - 2iT - 11T^{2} \) |
| 13 | \( 1 - 2.82iT - 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 5.65iT - 19T^{2} \) |
| 23 | \( 1 - 1.17T + 23T^{2} \) |
| 29 | \( 1 - 0.585iT - 29T^{2} \) |
| 31 | \( 1 + 4.58T + 31T^{2} \) |
| 37 | \( 1 - 9.65iT - 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 - 1.65iT - 43T^{2} \) |
| 47 | \( 1 + 12.4T + 47T^{2} \) |
| 53 | \( 1 + 11.8iT - 53T^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + 9.65iT - 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 - 9.17T + 71T^{2} \) |
| 73 | \( 1 - 1.65T + 73T^{2} \) |
| 79 | \( 1 + 5.75T + 79T^{2} \) |
| 83 | \( 1 + 9.31iT - 83T^{2} \) |
| 89 | \( 1 + 2T + 89T^{2} \) |
| 97 | \( 1 - 13.3T + 97T^{2} \) |
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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
−9.428507348460495500270621796176, −8.392706219590013956582903458297, −7.961084945334124835116899710559, −7.15175392242805359235699086887, −6.26121947452536303248036694556, −5.21257076605588841353768201564, −4.60898812251931788802516369023, −3.42297991747308763712519065650, −1.98320913006221134631679663812, −1.29437511196760295176668582230,
0.930560314517740518581942210336, 2.48840206132413182436529400434, 3.42021298445737023451488307758, 4.52658214244835027700213485213, 5.25776965184202928497629176289, 5.96431332839633585578637011046, 7.25618054984134027760977711487, 7.86151660873388503905290516826, 8.735677010310680927616748268775, 9.335511030231495265525307500885
