L(s) = 1 | + (1.32 − 1.11i)3-s − 2·4-s − 2.23i·5-s + 2.64·7-s + (0.5 − 2.95i)9-s + 5.91i·11-s + (−2.64 + 2.23i)12-s − 2.64·13-s + (−2.50 − 2.95i)15-s + 4·16-s + 2.23i·17-s + 4.47i·20-s + (3.50 − 2.95i)21-s − 5.00·25-s + (−2.64 − 4.47i)27-s − 5.29·28-s + ⋯ |
L(s) = 1 | + (0.763 − 0.645i)3-s − 4-s − 0.999i·5-s + 0.999·7-s + (0.166 − 0.986i)9-s + 1.78i·11-s + (−0.763 + 0.645i)12-s − 0.733·13-s + (−0.645 − 0.763i)15-s + 16-s + 0.542i·17-s + 0.999i·20-s + (0.763 − 0.645i)21-s − 1.00·25-s + (−0.509 − 0.860i)27-s − 0.999·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.645 + 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00584 - 0.466866i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00584 - 0.466866i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.32 + 1.11i)T \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 - 2.64T \) |
good | 2 | \( 1 + 2T^{2} \) |
| 11 | \( 1 - 5.91iT - 11T^{2} \) |
| 13 | \( 1 + 2.64T + 13T^{2} \) |
| 17 | \( 1 - 2.23iT - 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.91iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 11.1iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 + T + 79T^{2} \) |
| 83 | \( 1 - 8.94iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 18.5T + 97T^{2} \) |
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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
−13.56111116105316095342879797072, −12.60276359919700011842010315519, −12.12044609578012392963001748543, −10.04464913442371129729613491573, −9.109313852922315099750217969886, −8.236810291145322659413230385922, −7.30038568161409091706010431392, −5.15802979285512709996728673531, −4.21850871250961815429634595116, −1.72878415582342817772241598425,
2.94173244415565854990122695373, 4.28165375687532667794677697703, 5.62383035160571644093125849253, 7.66739092545290842768189535246, 8.489223512008288469056209217900, 9.579282568493256666071670355384, 10.64054151915667530332028350201, 11.54997381561968941826456409154, 13.38468222569331311466803811607, 14.14559230509448628332929173328