L(s) = 1 | + (0.835 + 1.14i)2-s + (1.10 − 1.33i)3-s + (−0.602 + 1.90i)4-s + (1 − 2i)5-s + (2.44 + 0.140i)6-s + 7-s + (−2.67 + 0.907i)8-s + (−0.569 − 2.94i)9-s + (3.11 − 0.531i)10-s + 2.20·11-s + (1.88 + 2.90i)12-s − 1.89i·13-s + (0.835 + 1.14i)14-s + (−1.56 − 3.54i)15-s + (−3.27 − 2.29i)16-s + 1.13·17-s + ⋯ |
L(s) = 1 | + (0.591 + 0.806i)2-s + (0.636 − 0.771i)3-s + (−0.301 + 0.953i)4-s + (0.447 − 0.894i)5-s + (0.998 + 0.0574i)6-s + 0.377·7-s + (−0.947 + 0.320i)8-s + (−0.189 − 0.981i)9-s + (0.985 − 0.167i)10-s + 0.664·11-s + (0.543 + 0.839i)12-s − 0.524i·13-s + (0.223 + 0.304i)14-s + (−0.405 − 0.914i)15-s + (−0.818 − 0.574i)16-s + 0.276·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.111i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.37537 + 0.132337i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.37537 + 0.132337i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.835 - 1.14i)T \) |
| 3 | \( 1 + (-1.10 + 1.33i)T \) |
| 5 | \( 1 + (-1 + 2i)T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 2.20T + 11T^{2} \) |
| 13 | \( 1 + 1.89iT - 13T^{2} \) |
| 17 | \( 1 - 1.13T + 17T^{2} \) |
| 19 | \( 1 - 8.56iT - 19T^{2} \) |
| 23 | \( 1 + 3.21iT - 23T^{2} \) |
| 29 | \( 1 - 1.89iT - 29T^{2} \) |
| 31 | \( 1 - 5.90iT - 31T^{2} \) |
| 37 | \( 1 - 0.409iT - 37T^{2} \) |
| 41 | \( 1 - 4.40iT - 41T^{2} \) |
| 43 | \( 1 + 0.934T + 43T^{2} \) |
| 47 | \( 1 - 2.67iT - 47T^{2} \) |
| 53 | \( 1 + 6.81T + 53T^{2} \) |
| 59 | \( 1 + 13.5T + 59T^{2} \) |
| 61 | \( 1 - 12.4T + 61T^{2} \) |
| 67 | \( 1 + 10.8T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 3.76iT - 79T^{2} \) |
| 83 | \( 1 + 6.84iT - 83T^{2} \) |
| 89 | \( 1 - 16.1iT - 89T^{2} \) |
| 97 | \( 1 + 18.4iT - 97T^{2} \) |
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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
−11.73661535659481206275624809490, −10.04110852281352022947695730873, −8.955216836775243830040107474914, −8.309390418075724492866591321169, −7.61103738340364389575417072897, −6.40091357015312444951906726912, −5.66325777951156073206922414393, −4.43315845006297842323589090967, −3.23749000233078838930528540860, −1.54898627994113520815872536121,
2.01413902754149378612222467916, 3.03597880506167268154064148991, 4.08744286755495355888268657025, 5.07436612849118233598200416153, 6.25418128538615455074961187735, 7.41910979193041177654448983771, 8.981225597799270489321245219642, 9.469456981305286299950115294100, 10.38310199823424405902618039675, 11.20822423743695479333261995172