L(s) = 1 | + (−0.779 + 1.54i)3-s + (−0.268 + 2.21i)5-s + 2.49·7-s + (−1.78 − 2.41i)9-s − 3.17·11-s − 5.31i·13-s + (−3.22 − 2.14i)15-s − 2.70i·17-s − 8.13i·19-s + (−1.94 + 3.86i)21-s + (−4.12 − 2.44i)23-s + (−4.85 − 1.19i)25-s + (5.12 − 0.877i)27-s + 2.82i·29-s − 6.28·31-s + ⋯ |
L(s) = 1 | + (−0.450 + 0.892i)3-s + (−0.119 + 0.992i)5-s + 0.944·7-s + (−0.594 − 0.804i)9-s − 0.958·11-s − 1.47i·13-s + (−0.832 − 0.554i)15-s − 0.655i·17-s − 1.86i·19-s + (−0.425 + 0.843i)21-s + (−0.859 − 0.510i)23-s + (−0.971 − 0.238i)25-s + (0.985 − 0.168i)27-s + 0.524i·29-s − 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.432 + 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7340628391\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7340628391\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.779 - 1.54i)T \) |
| 5 | \( 1 + (0.268 - 2.21i)T \) |
| 23 | \( 1 + (4.12 + 2.44i)T \) |
good | 7 | \( 1 - 2.49T + 7T^{2} \) |
| 11 | \( 1 + 3.17T + 11T^{2} \) |
| 13 | \( 1 + 5.31iT - 13T^{2} \) |
| 17 | \( 1 + 2.70iT - 17T^{2} \) |
| 19 | \( 1 + 8.13iT - 19T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 6.28T + 31T^{2} \) |
| 37 | \( 1 - 3.82T + 37T^{2} \) |
| 41 | \( 1 + 3.17iT - 41T^{2} \) |
| 43 | \( 1 - 6.93T + 43T^{2} \) |
| 47 | \( 1 + 8.10T + 47T^{2} \) |
| 53 | \( 1 + 3.79iT - 53T^{2} \) |
| 59 | \( 1 - 11.2iT - 59T^{2} \) |
| 61 | \( 1 - 0.443iT - 61T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 + 14.4iT - 71T^{2} \) |
| 73 | \( 1 - 2.80iT - 73T^{2} \) |
| 79 | \( 1 - 6.08iT - 79T^{2} \) |
| 83 | \( 1 - 8.29iT - 83T^{2} \) |
| 89 | \( 1 + 9.80T + 89T^{2} \) |
| 97 | \( 1 - 16.6T + 97T^{2} \) |
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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
−9.641562690789993023266126021012, −8.638216121984359054390240908176, −7.78454410353188824445927549288, −7.07296109553288805583034752918, −5.91097565680939411673527685831, −5.20625347984175195855071885892, −4.47261108560258673777659960001, −3.22442103003039814107005968042, −2.51380526219318837879340220093, −0.31058409139886734360071590945,
1.49878454119805496171260354369, 2.01312596704538871220185963096, 3.90580155096929066579300876069, 4.78534030835498489675868201761, 5.63571772405778845009516532074, 6.27404646295130297023521555719, 7.69722859543548496210089853550, 7.86228359096577921235878607956, 8.674739642448917145814739004224, 9.681288681099615924100403716223