L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.587 − 1.80i)3-s + (0.309 + 0.951i)4-s + (−0.587 + 0.809i)5-s + (0.587 − 1.80i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−2.11 + 1.53i)9-s + (−0.951 + 0.309i)10-s + (1.53 − 1.11i)12-s + (−1.53 + 1.11i)13-s + (−0.809 − 0.587i)14-s + (1.80 + 0.587i)15-s + (−0.809 + 0.587i)16-s − 2.61·18-s + ⋯ |
L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.587 − 1.80i)3-s + (0.309 + 0.951i)4-s + (−0.587 + 0.809i)5-s + (0.587 − 1.80i)6-s − 7-s + (−0.309 + 0.951i)8-s + (−2.11 + 1.53i)9-s + (−0.951 + 0.309i)10-s + (1.53 − 1.11i)12-s + (−1.53 + 1.11i)13-s + (−0.809 − 0.587i)14-s + (1.80 + 0.587i)15-s + (−0.809 + 0.587i)16-s − 2.61·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4207377035\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4207377035\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (0.587 - 0.809i)T \) |
| 7 | \( 1 + T \) |
good | 3 | \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \) |
| 11 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \) |
| 17 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 53 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 59 | \( 1 + (0.951 - 0.690i)T + (0.309 - 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 71 | \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \) |
| 73 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 79 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 97 | \( 1 + (0.809 - 0.587i)T^{2} \) |
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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
−10.24301713716002207965098355252, −8.930164656157059179863126857676, −7.86611207059871618923586164257, −7.32097094007571088730854718975, −6.74590527902205714897938412729, −6.33663748713288865816398516465, −5.38039590748610933283239612216, −4.19190532648003138063654156884, −2.87030431347553933941278850483, −2.22580581041697133799386198089,
0.26003480689122438151243226334, 2.80773861137065744452294652644, 3.59567777942205648497474576155, 4.31235269720788028879397909266, 5.17308502305475015503440152735, 5.53231800966955359884343061761, 6.62229065319014939290435191279, 7.966693095700934879688907089237, 9.277510884324734512567284253942, 9.602565243220173100161443657358