| L(s) = 1 | + (−0.707 + 1.22i)2-s + (−0.999 − 1.73i)4-s + 3.46i·7-s + 2.82·8-s − 3.16·11-s − 4.47·13-s + (−4.24 − 2.44i)14-s + (−2.00 + 3.46i)16-s + 6.32·17-s + (−2 + 3.87i)19-s + (2.23 − 3.87i)22-s − 5.47i·23-s + 5·25-s + (3.16 − 5.47i)26-s + (5.99 − 3.46i)28-s − 1.41·29-s + ⋯ |
| L(s) = 1 | + (−0.499 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 1.30i·7-s + 0.999·8-s − 0.953·11-s − 1.24·13-s + (−1.13 − 0.654i)14-s + (−0.500 + 0.866i)16-s + 1.53·17-s + (−0.458 + 0.888i)19-s + (0.476 − 0.825i)22-s − 1.14i·23-s + 25-s + (0.620 − 1.07i)26-s + (1.13 − 0.654i)28-s − 0.262·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.458 + 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1375792484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1375792484\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.707 - 1.22i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (2 - 3.87i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 17 | \( 1 - 6.32T + 17T^{2} \) |
| 23 | \( 1 + 5.47iT - 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 + 4.47T + 37T^{2} \) |
| 41 | \( 1 + 2.44iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 5.47iT - 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 4.89iT - 59T^{2} \) |
| 61 | \( 1 - 10.3iT - 61T^{2} \) |
| 67 | \( 1 + 7.74iT - 67T^{2} \) |
| 71 | \( 1 + 11.3T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 17.1iT - 89T^{2} \) |
| 97 | \( 1 + 15.4iT - 97T^{2} \) |
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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
−10.22409483794360176983390186280, −9.057066057338843096480148910479, −8.575201771524303087726439190524, −7.67774621087090181239300713860, −7.08453345304653042443763017224, −5.78281497783461297013428681732, −5.50663766691600350576572161999, −4.57713903706899231244780099313, −2.98896350510321427867348845745, −1.86995344646743254585171739210,
0.06695673624311332273127829148, 1.41349188524016326130338442361, 2.77176384375989408696164551248, 3.58638793025680342308080995334, 4.65952944633862404874550341770, 5.39381039848759633932571201323, 7.17134534400544727721667221502, 7.39518247873611867881247747647, 8.218978421673654360647973512445, 9.360480584994099887901037688114
