| L(s) = 1 | − i·3-s + (−2 + i)5-s − 9-s + 4·11-s + 2i·13-s + (1 + 2i)15-s − 2i·17-s − 2·19-s + 6i·23-s + (3 − 4i)25-s + i·27-s − 6·29-s − 6·31-s − 4i·33-s − 4i·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.894 + 0.447i)5-s − 0.333·9-s + 1.20·11-s + 0.554i·13-s + (0.258 + 0.516i)15-s − 0.485i·17-s − 0.458·19-s + 1.25i·23-s + (0.600 − 0.800i)25-s + 0.192i·27-s − 1.11·29-s − 1.07·31-s − 0.696i·33-s − 0.657i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6331447403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6331447403\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 + (2 - i)T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 6T + 31T^{2} \) |
| 37 | \( 1 + 4iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 4iT - 47T^{2} \) |
| 53 | \( 1 - 2iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 - 16T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.051048947794628533510196186296, −8.106278519443134219659022432091, −7.35272210053040243215126181516, −6.92294673127193780284886770338, −6.13936801617107688596707565968, −5.18845594879441529510928571338, −4.00034582203185742165614556340, −3.59885183517094469271281666423, −2.35578237632128651592227694900, −1.28675044135818255554348063762,
0.21196124680809449861898355733, 1.60714240219391453723133284135, 3.03998099555533819756097299373, 3.93053625747555283213109543085, 4.35894818581600600722463576798, 5.33834232676196995198762371115, 6.18011343380838597022912066297, 7.06090559064740759327555742968, 7.85114431105295662997473218708, 8.756325244399062853672079823269
