| L(s) = 1 | + i·3-s + 0.732i·5-s + 7-s − 9-s + 1.26i·11-s + 5.46i·13-s − 0.732·15-s − 4.19·17-s − 0.535i·19-s + i·21-s + 3.26·23-s + 4.46·25-s − i·27-s + 3.46i·29-s − 2·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.327i·5-s + 0.377·7-s − 0.333·9-s + 0.382i·11-s + 1.51i·13-s − 0.189·15-s − 1.01·17-s − 0.122i·19-s + 0.218i·21-s + 0.681·23-s + 0.892·25-s − 0.192i·27-s + 0.643i·29-s − 0.359·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.811412 + 1.05745i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.811412 + 1.05745i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 - 0.732iT - 5T^{2} \) |
| 11 | \( 1 - 1.26iT - 11T^{2} \) |
| 13 | \( 1 - 5.46iT - 13T^{2} \) |
| 17 | \( 1 + 4.19T + 17T^{2} \) |
| 19 | \( 1 + 0.535iT - 19T^{2} \) |
| 23 | \( 1 - 3.26T + 23T^{2} \) |
| 29 | \( 1 - 3.46iT - 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 8.92iT - 37T^{2} \) |
| 41 | \( 1 + 9.66T + 41T^{2} \) |
| 43 | \( 1 - 7.46iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 14.9iT - 59T^{2} \) |
| 61 | \( 1 + 6.92iT - 61T^{2} \) |
| 67 | \( 1 + 6iT - 67T^{2} \) |
| 71 | \( 1 + 7.26T + 71T^{2} \) |
| 73 | \( 1 - 15.4T + 73T^{2} \) |
| 79 | \( 1 - 12.3T + 79T^{2} \) |
| 83 | \( 1 + 1.46iT - 83T^{2} \) |
| 89 | \( 1 - 6.73T + 89T^{2} \) |
| 97 | \( 1 + 4.53T + 97T^{2} \) |
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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
−10.87385171694592293296401195070, −9.858088579659719051113970570919, −9.071410326285233454573930480092, −8.373207563877016072684889060545, −7.01969073645720311612967844410, −6.51979976680472075725341591515, −5.01238559126015744436922975078, −4.44171180471367354807683454683, −3.18083657379653966542518457480, −1.83789826925857281398348692097,
0.72040254140763017645693044492, 2.27038949755171512293374957717, 3.50704449651644137770354065918, 4.91145292875108467442409102525, 5.69279163159107495288387391420, 6.76621368751419490871232232448, 7.69553570883264387635495102050, 8.474914603931418842417198191066, 9.169133722923300492357322001917, 10.49078255514813676200298900352
