| L(s) = 1 | + i·3-s − i·5-s − 2·7-s − 9-s − 2i·11-s + 2i·13-s + 15-s + 4·17-s + 4i·19-s − 2i·21-s − 4·23-s − 25-s − i·27-s + 2i·29-s − 4·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.447i·5-s − 0.755·7-s − 0.333·9-s − 0.603i·11-s + 0.554i·13-s + 0.258·15-s + 0.970·17-s + 0.917i·19-s − 0.436i·21-s − 0.834·23-s − 0.200·25-s − 0.192i·27-s + 0.371i·29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7888945846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7888945846\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + iT \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 4T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 - 10iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 16iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14T + 97T^{2} \) |
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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
−9.641214052552447118854817027544, −8.767485436494887063419299969576, −8.091336129868759217900724226901, −7.17606597733154616563819357780, −6.03297402500341719169067061997, −5.66396393363963465040847778028, −4.48796454073255759835434882804, −3.71639244827035964013692630335, −2.88724812615568754402538124865, −1.38644709173511764815421094461,
0.28752645143261348999817777872, 1.86768577791447890582072157859, 2.91086673768889109528669180180, 3.69469712571173470165046296228, 4.94519115195961000641508632514, 5.89309596563536002351928267567, 6.55596640815945222061463406359, 7.37914790000760467276501922256, 7.899202937869052609940500356239, 8.937392507723227211213638940724
