| L(s) = 1 | + i·3-s + 2i·5-s − 7-s − 9-s − 4i·11-s + 6i·13-s − 2·15-s + 2·17-s + 4i·19-s − i·21-s + 8·23-s + 25-s − i·27-s − 2i·29-s + 4·33-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.894i·5-s − 0.377·7-s − 0.333·9-s − 1.20i·11-s + 1.66i·13-s − 0.516·15-s + 0.485·17-s + 0.917i·19-s − 0.218i·21-s + 1.66·23-s + 0.200·25-s − 0.192i·27-s − 0.371i·29-s + 0.696·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.672647059\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.672647059\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2iT - 5T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + 2iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 - 4iT - 59T^{2} \) |
| 61 | \( 1 - 6iT - 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 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
−8.621366452019190887320631839026, −7.70517792989480309540567089013, −6.83002515868643945647206845735, −6.40217133796447222721271429009, −5.63144174711826700404796348786, −4.75110138175442622957968160616, −3.83309676432338937692160833564, −3.24915790099916601773208035155, −2.51470438863469868490803006520, −1.16587748387176561610180122599,
0.51055117947276494161603021739, 1.28471097913932573561104526028, 2.53036354319024273378236908814, 3.18614034514661088383140031070, 4.36099879761667728832218104272, 5.13678929617150251258547586216, 5.57126340951398964941964275979, 6.59696436276526847162265099095, 7.33400089519323331662822113035, 7.74431598740842769535414443246
