L(s) = 1 | − 0.615i·2-s + i·3-s + 1.62·4-s − 3.28i·5-s + 0.615·6-s − 2.22i·8-s − 9-s − 2.02·10-s + (−0.0656 − 3.31i)11-s + 1.62i·12-s + 0.142·13-s + 3.28·15-s + 1.87·16-s − 7.26·17-s + 0.615i·18-s − 0.869·19-s + ⋯ |
L(s) = 1 | − 0.435i·2-s + 0.577i·3-s + 0.810·4-s − 1.46i·5-s + 0.251·6-s − 0.788i·8-s − 0.333·9-s − 0.639·10-s + (−0.0197 − 0.999i)11-s + 0.467i·12-s + 0.0395·13-s + 0.848·15-s + 0.467·16-s − 1.76·17-s + 0.145i·18-s − 0.199·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.742 + 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.569604688\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.569604688\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (0.0656 + 3.31i)T \) |
good | 2 | \( 1 + 0.615iT - 2T^{2} \) |
| 5 | \( 1 + 3.28iT - 5T^{2} \) |
| 13 | \( 1 - 0.142T + 13T^{2} \) |
| 17 | \( 1 + 7.26T + 17T^{2} \) |
| 19 | \( 1 + 0.869T + 19T^{2} \) |
| 23 | \( 1 + 3.56T + 23T^{2} \) |
| 29 | \( 1 + 3.70iT - 29T^{2} \) |
| 31 | \( 1 + 4.82iT - 31T^{2} \) |
| 37 | \( 1 - 9.39T + 37T^{2} \) |
| 41 | \( 1 + 3.19T + 41T^{2} \) |
| 43 | \( 1 - 9.66iT - 43T^{2} \) |
| 47 | \( 1 - 8.76iT - 47T^{2} \) |
| 53 | \( 1 + 6.62T + 53T^{2} \) |
| 59 | \( 1 + 1.54iT - 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 - 2.66T + 71T^{2} \) |
| 73 | \( 1 - 5.48T + 73T^{2} \) |
| 79 | \( 1 + 7.05iT - 79T^{2} \) |
| 83 | \( 1 - 6.31T + 83T^{2} \) |
| 89 | \( 1 + 7.87iT - 89T^{2} \) |
| 97 | \( 1 + 8.07iT - 97T^{2} \) |
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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
−9.282830741484887341335305861698, −8.352655927923652850197433206390, −7.80548096113089553089518765730, −6.33461615263627596566987324660, −5.94676465723535935371361110276, −4.67243508115196179307364389284, −4.15111420768180260817261431133, −2.94804382213526458578929036902, −1.86209607102753946022519353587, −0.54165649687048734556918312696,
1.98279274601589004691109909012, 2.45396594304447504459314499807, 3.58193908935762236079862391306, 4.89417607736762393988751461599, 6.09007472654165641244643396858, 6.71500536742912170735681024287, 7.03879739701040112753651105786, 7.79526710689230100358632201310, 8.684356143011238016913252626758, 9.811553062138127586684992959386