L(s) = 1 | + (1.05 − 1.82i)5-s + (−1.43 − 2.49i)7-s + (1.21 + 2.10i)11-s + (3.30 − 5.71i)13-s + 7.56·17-s + 6.25·19-s + (−2.63 + 4.56i)23-s + (0.275 + 0.476i)25-s + (1.57 + 2.73i)29-s + (1.79 − 3.10i)31-s − 6.07·35-s − 6.70·37-s + (1.74 − 3.02i)41-s + (3.12 + 5.41i)43-s + (1.32 + 2.30i)47-s + ⋯ |
L(s) = 1 | + (0.471 − 0.816i)5-s + (−0.543 − 0.942i)7-s + (0.365 + 0.633i)11-s + (0.915 − 1.58i)13-s + 1.83·17-s + 1.43·19-s + (−0.549 + 0.952i)23-s + (0.0550 + 0.0953i)25-s + (0.293 + 0.507i)29-s + (0.321 − 0.556i)31-s − 1.02·35-s − 1.10·37-s + (0.272 − 0.472i)41-s + (0.476 + 0.825i)43-s + (0.193 + 0.335i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.332471128\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.332471128\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.05 + 1.82i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (1.43 + 2.49i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.21 - 2.10i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.30 + 5.71i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 7.56T + 17T^{2} \) |
| 19 | \( 1 - 6.25T + 19T^{2} \) |
| 23 | \( 1 + (2.63 - 4.56i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.57 - 2.73i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.79 + 3.10i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.70T + 37T^{2} \) |
| 41 | \( 1 + (-1.74 + 3.02i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.12 - 5.41i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.32 - 2.30i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 0.953T + 53T^{2} \) |
| 59 | \( 1 + (4.84 - 8.39i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.57 - 4.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.949 + 1.64i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 5.82T + 71T^{2} \) |
| 73 | \( 1 + 5.01T + 73T^{2} \) |
| 79 | \( 1 + (6.49 + 11.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.54 + 2.67i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.95T + 89T^{2} \) |
| 97 | \( 1 + (5.51 + 9.54i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.424134971393613391790983238049, −7.55574276239703018904262055944, −7.26426501429666136434616134853, −5.83364457540739662185995253744, −5.66211623566214419559169643290, −4.68501711098075425602637417286, −3.55373947564961170155658462771, −3.17116552220397500480540298325, −1.39805559204521092093884337549, −0.892605079808426276219323521603,
1.17911881230769671766816772585, 2.33127462883853417175066959254, 3.19540187531946204595804472771, 3.81675510833696553418828051130, 5.12115258926327177469752171024, 5.93993953452292762902722760835, 6.37375634375456526021885305201, 7.05135783943462477746733714584, 8.088061471976752399032574085233, 8.805326511540337329077128051295