| L(s) = 1 | + 0.873i·3-s + (−1.89 − 1.18i)5-s + 0.992i·7-s + 2.23·9-s + 1.83·11-s + 3.28i·13-s + (1.03 − 1.65i)15-s + 6.63i·17-s + 5.64·19-s − 0.866·21-s − i·23-s + (2.20 + 4.48i)25-s + 4.57i·27-s − 2.01·29-s − 0.315·31-s + ⋯ |
| L(s) = 1 | + 0.504i·3-s + (−0.848 − 0.528i)5-s + 0.375i·7-s + 0.745·9-s + 0.552·11-s + 0.911i·13-s + (0.266 − 0.428i)15-s + 1.60i·17-s + 1.29·19-s − 0.189·21-s − 0.208i·23-s + (0.441 + 0.897i)25-s + 0.880i·27-s − 0.374·29-s − 0.0565·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12429 + 0.624414i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12429 + 0.624414i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.89 + 1.18i)T \) |
| 23 | \( 1 + iT \) |
| good | 3 | \( 1 - 0.873iT - 3T^{2} \) |
| 7 | \( 1 - 0.992iT - 7T^{2} \) |
| 11 | \( 1 - 1.83T + 11T^{2} \) |
| 13 | \( 1 - 3.28iT - 13T^{2} \) |
| 17 | \( 1 - 6.63iT - 17T^{2} \) |
| 19 | \( 1 - 5.64T + 19T^{2} \) |
| 29 | \( 1 + 2.01T + 29T^{2} \) |
| 31 | \( 1 + 0.315T + 31T^{2} \) |
| 37 | \( 1 - 3.07iT - 37T^{2} \) |
| 41 | \( 1 + 1.34T + 41T^{2} \) |
| 43 | \( 1 + 5.97iT - 43T^{2} \) |
| 47 | \( 1 + 0.306iT - 47T^{2} \) |
| 53 | \( 1 + 6.98iT - 53T^{2} \) |
| 59 | \( 1 + 9.49T + 59T^{2} \) |
| 61 | \( 1 - 5.56T + 61T^{2} \) |
| 67 | \( 1 + 0.853iT - 67T^{2} \) |
| 71 | \( 1 - 0.797T + 71T^{2} \) |
| 73 | \( 1 - 7.67iT - 73T^{2} \) |
| 79 | \( 1 - 3.62T + 79T^{2} \) |
| 83 | \( 1 + 17.1iT - 83T^{2} \) |
| 89 | \( 1 - 7.01T + 89T^{2} \) |
| 97 | \( 1 - 18.5iT - 97T^{2} \) |
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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
−11.28710781983630650637863347108, −10.26638239315693069261643807036, −9.328358097065452911722007211620, −8.639802287953400729939093114437, −7.60516290674688592053100301283, −6.63100425435688680089206144224, −5.32597429041428427516185258908, −4.26707956231296707206977121027, −3.59248370712056188033603406946, −1.55425009270289474880668409007,
0.926402830698876198331073085103, 2.85386729964126396438061054299, 3.92798921790456463879258815914, 5.12214250300049036943350075276, 6.52286891780431709521285400724, 7.47358579123453851202438304958, 7.68782855530926734475563314398, 9.184975060526555974769966533952, 10.05337120260995481293317132025, 11.03818897054296736409745129360
