L(s) = 1 | + 2.99i·5-s − 7-s − 2.13i·11-s − 0.665i·13-s − 7.04·17-s − 1.39i·19-s + 0.184·23-s − 3.94·25-s + 1.27i·29-s + 7.62·31-s − 2.99i·35-s + 6.69i·37-s + 0.274·41-s + 2.63i·43-s − 5.77·47-s + ⋯ |
L(s) = 1 | + 1.33i·5-s − 0.377·7-s − 0.645i·11-s − 0.184i·13-s − 1.70·17-s − 0.320i·19-s + 0.0384·23-s − 0.788·25-s + 0.236i·29-s + 1.36·31-s − 0.505i·35-s + 1.10i·37-s + 0.0428·41-s + 0.402i·43-s − 0.842·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.076458196\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.076458196\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 - 2.99iT - 5T^{2} \) |
| 11 | \( 1 + 2.13iT - 11T^{2} \) |
| 13 | \( 1 + 0.665iT - 13T^{2} \) |
| 17 | \( 1 + 7.04T + 17T^{2} \) |
| 19 | \( 1 + 1.39iT - 19T^{2} \) |
| 23 | \( 1 - 0.184T + 23T^{2} \) |
| 29 | \( 1 - 1.27iT - 29T^{2} \) |
| 31 | \( 1 - 7.62T + 31T^{2} \) |
| 37 | \( 1 - 6.69iT - 37T^{2} \) |
| 41 | \( 1 - 0.274T + 41T^{2} \) |
| 43 | \( 1 - 2.63iT - 43T^{2} \) |
| 47 | \( 1 + 5.77T + 47T^{2} \) |
| 53 | \( 1 + 7.48iT - 53T^{2} \) |
| 59 | \( 1 + 9.03iT - 59T^{2} \) |
| 61 | \( 1 + 9.80iT - 61T^{2} \) |
| 67 | \( 1 + 10.8iT - 67T^{2} \) |
| 71 | \( 1 - 7.93T + 71T^{2} \) |
| 73 | \( 1 + 2.80T + 73T^{2} \) |
| 79 | \( 1 + 7.98T + 79T^{2} \) |
| 83 | \( 1 - 6.76iT - 83T^{2} \) |
| 89 | \( 1 + 16.7T + 89T^{2} \) |
| 97 | \( 1 + 0.107T + 97T^{2} \) |
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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
−8.070812519712291687336774521234, −6.96491110036188830044117060166, −6.62283803375149706464890016894, −6.15771819766520238973512245869, −5.06866641586695547547907603608, −4.28350392871463377282347002819, −3.24550969050668296566540516173, −2.86569521845771541064653433314, −1.89529099931313700663338453438, −0.32235762301354009281713925685,
0.894234483912562051179571043720, 1.92932394663155921621564397087, 2.77748397368066467123686594195, 4.20794448922653639888849555763, 4.33822088886532796865711107708, 5.24607741144970297110282251183, 6.00647252457115133187145465321, 6.77768581567085229313155690569, 7.45695703894193856053412920067, 8.375822944062922134788917511136