| L(s) = 1 | − 2i·5-s + 4i·11-s − 2i·13-s − 2·17-s − 4i·19-s − 8·23-s + 25-s − 6i·29-s + 8·31-s − 6i·37-s − 6·41-s − 4i·43-s − 7·49-s − 2i·53-s + 8·55-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s + 1.20i·11-s − 0.554i·13-s − 0.485·17-s − 0.917i·19-s − 1.66·23-s + 0.200·25-s − 1.11i·29-s + 1.43·31-s − 0.986i·37-s − 0.937·41-s − 0.609i·43-s − 49-s − 0.274i·53-s + 1.07·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9732684211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9732684211\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 - 4iT - 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + 2T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.607220065156048829403997413640, −8.094825131228622968700602592412, −7.20838689010361996648963147661, −6.39646513753830288613233671949, −5.45088594781412998577898714603, −4.63388429120213377479475858963, −4.09953842333755903220408327640, −2.69099411753690930833341423639, −1.74000517821645087039298278051, −0.32496190451294687638815284237,
1.47608256025237012465449908004, 2.71995460686128039692108259734, 3.46762236550596602135600270505, 4.37885171867923715420089821869, 5.48503749812980556220380432607, 6.42217389574043078852694510307, 6.68218875941622221512850966157, 7.938371580462713834192878948760, 8.356040830883049492056128192654, 9.293519915764770492117794630436
