| L(s) = 1 | − i·3-s − 3i·7-s − 9-s + 2·11-s − 3i·13-s + 6i·17-s + 7·19-s − 3·21-s + 6i·23-s + i·27-s − 2·29-s + 5·31-s − 2i·33-s − 10i·37-s − 3·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.13i·7-s − 0.333·9-s + 0.603·11-s − 0.832i·13-s + 1.45i·17-s + 1.60·19-s − 0.654·21-s + 1.25i·23-s + 0.192i·27-s − 0.371·29-s + 0.898·31-s − 0.348i·33-s − 1.64i·37-s − 0.480·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.155247061\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.155247061\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 + 3iT - 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 7T + 19T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 + 3iT - 43T^{2} \) |
| 47 | \( 1 - 10iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 13T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 - 6iT - 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 + 16T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−7.938221310560828292654243092291, −7.43140856264504493673792403403, −6.93174011971981996490514855232, −5.84185834318970742589768883747, −5.53250771010055608889464874269, −4.16798076828796999795368539329, −3.72202399456383203643213891536, −2.73845633439929779830144635468, −1.45423069952354415473069150467, −0.809929797341450769054964834201,
0.929107553801735562031206855379, 2.34322944440169654584417784809, 2.93671936319750153627805690969, 3.94374071108469997562821317102, 4.84443366446912856076163666733, 5.30571021535095644880219082614, 6.24483117204824828233149814307, 6.85219791896510048477784859271, 7.74695374493879989485486476684, 8.631404241044679861006796995703
