| L(s) = 1 | − i·3-s − i·7-s − 9-s − 11-s + 2i·13-s + 8i·17-s + 2·19-s − 21-s − i·23-s + i·27-s − 29-s + 6·31-s + i·33-s + 9i·37-s + 2·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.377i·7-s − 0.333·9-s − 0.301·11-s + 0.554i·13-s + 1.94i·17-s + 0.458·19-s − 0.218·21-s − 0.208i·23-s + 0.192i·27-s − 0.185·29-s + 1.07·31-s + 0.174i·33-s + 1.47i·37-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.516636078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.516636078\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 11 | \( 1 + T + 11T^{2} \) |
| 13 | \( 1 - 2iT - 13T^{2} \) |
| 17 | \( 1 - 8iT - 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 + iT - 23T^{2} \) |
| 29 | \( 1 + T + 29T^{2} \) |
| 31 | \( 1 - 6T + 31T^{2} \) |
| 37 | \( 1 - 9iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - iT - 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 6T + 59T^{2} \) |
| 61 | \( 1 - 8T + 61T^{2} \) |
| 67 | \( 1 + 3iT - 67T^{2} \) |
| 71 | \( 1 - 7T + 71T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 - 14iT - 97T^{2} \) |
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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
−9.056106742835167612208737678560, −8.180951193351855861237106914913, −7.78761516803061992140063303776, −6.64317519577752654079971723731, −6.28433790545918520479219649234, −5.19864729965188983220432715683, −4.24340029188606645835860949655, −3.32685085503430535779699711367, −2.14846930701554384955891958005, −1.12859972653374871501678027355,
0.61147813909805534924531335199, 2.38464777795856226612804372072, 3.12037019551736902702386465873, 4.17580076043893939313622680605, 5.22208162987546580381725167446, 5.53666102022572812183341739802, 6.77464435484623852628988919162, 7.51459228744039488110256491092, 8.358062888968372300976287493848, 9.159402334906674063456710836325
