| L(s) = 1 | + (1.22 + 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 + 1.73i)4-s + 2.44i·6-s + (2.86 − 6.38i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (−9.98 − 17.2i)11-s + (−1.73 + 2.99i)12-s − 3.49·13-s + (8.02 − 5.80i)14-s + (−2.00 + 3.46i)16-s + (9.12 + 15.7i)17-s + (−3.67 + 2.12i)18-s + (21.3 + 12.3i)19-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (0.288 + 0.5i)3-s + (0.249 + 0.433i)4-s + 0.408i·6-s + (0.408 − 0.912i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.907 − 1.57i)11-s + (−0.144 + 0.249i)12-s − 0.269·13-s + (0.572 − 0.414i)14-s + (−0.125 + 0.216i)16-s + (0.536 + 0.929i)17-s + (−0.204 + 0.117i)18-s + (1.12 + 0.647i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.847 - 0.530i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.253254645\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.253254645\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (-0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.86 + 6.38i)T \) |
| good | 11 | \( 1 + (9.98 + 17.2i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 3.49T + 169T^{2} \) |
| 17 | \( 1 + (-9.12 - 15.7i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-21.3 - 12.3i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-21.8 - 12.5i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 53.1T + 841T^{2} \) |
| 31 | \( 1 + (-26.0 + 15.0i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-40.5 - 23.3i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 31.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 64.4iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-14.0 + 24.3i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-56.1 + 32.4i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (86.7 - 50.0i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-6.94 - 4.01i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-14.0 + 8.13i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 107.T + 5.04e3T^{2} \) |
| 73 | \( 1 + (25.8 + 44.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (10.9 - 19.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 0.417T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-96.3 - 55.6i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 74.2T + 9.40e3T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05915411568721375202677289320, −8.698511635471899766996232107922, −8.051644959423957845090919671893, −7.44253581690896170292058115300, −6.20949787566174774078723011934, −5.41532674699707894776741896232, −4.57816612288908802059841648749, −3.53559290990090050446005333937, −2.87050338731856512627792843333, −1.00683514346438915311510987100,
1.10498836002760038932962353302, 2.56607244789170499790834106560, 2.80884010804755691674922087045, 4.75087731233427712282671655869, 4.96028253358974915634332643838, 6.18310770278231807001088391287, 7.23073234957400720714848909536, 7.75958684502507513705053254466, 8.933346668843456701509821447231, 9.669307847789920915773929222817
