L(s) = 1 | + (−1.13 − 0.847i)2-s + (0.707 − 0.707i)3-s + (0.563 + 1.91i)4-s + (0.707 + 0.707i)5-s + (−1.39 + 0.201i)6-s + 4.27i·7-s + (0.989 − 2.64i)8-s − 1.00i·9-s + (−0.201 − 1.39i)10-s + (2.94 + 2.94i)11-s + (1.75 + 0.958i)12-s + (−4.05 + 4.05i)13-s + (3.62 − 4.83i)14-s + 1.00·15-s + (−3.36 + 2.16i)16-s + 0.160·17-s + ⋯ |
L(s) = 1 | + (−0.800 − 0.599i)2-s + (0.408 − 0.408i)3-s + (0.281 + 0.959i)4-s + (0.316 + 0.316i)5-s + (−0.571 + 0.0821i)6-s + 1.61i·7-s + (0.349 − 0.936i)8-s − 0.333i·9-s + (−0.0636 − 0.442i)10-s + (0.887 + 0.887i)11-s + (0.506 + 0.276i)12-s + (−1.12 + 1.12i)13-s + (0.967 − 1.29i)14-s + 0.258·15-s + (−0.841 + 0.540i)16-s + 0.0388·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.177i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00795 + 0.0900325i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00795 + 0.0900325i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.13 + 0.847i)T \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
good | 7 | \( 1 - 4.27iT - 7T^{2} \) |
| 11 | \( 1 + (-2.94 - 2.94i)T + 11iT^{2} \) |
| 13 | \( 1 + (4.05 - 4.05i)T - 13iT^{2} \) |
| 17 | \( 1 - 0.160T + 17T^{2} \) |
| 19 | \( 1 + (-4.32 + 4.32i)T - 19iT^{2} \) |
| 23 | \( 1 + 8.40iT - 23T^{2} \) |
| 29 | \( 1 + (1.78 - 1.78i)T - 29iT^{2} \) |
| 31 | \( 1 - 7.17T + 31T^{2} \) |
| 37 | \( 1 + (0.669 + 0.669i)T + 37iT^{2} \) |
| 41 | \( 1 + 3.96iT - 41T^{2} \) |
| 43 | \( 1 + (0.255 + 0.255i)T + 43iT^{2} \) |
| 47 | \( 1 - 0.0752T + 47T^{2} \) |
| 53 | \( 1 + (2.88 + 2.88i)T + 53iT^{2} \) |
| 59 | \( 1 + (5.63 + 5.63i)T + 59iT^{2} \) |
| 61 | \( 1 + (4.48 - 4.48i)T - 61iT^{2} \) |
| 67 | \( 1 + (0.131 - 0.131i)T - 67iT^{2} \) |
| 71 | \( 1 - 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 0.382iT - 73T^{2} \) |
| 79 | \( 1 - 15.3T + 79T^{2} \) |
| 83 | \( 1 + (5.54 - 5.54i)T - 83iT^{2} \) |
| 89 | \( 1 + 13.8iT - 89T^{2} \) |
| 97 | \( 1 - 10.8T + 97T^{2} \) |
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
−12.12575994105587085356531068546, −11.41652412433458264667060987481, −9.891789026742433817662867901489, −9.267149831493328057929314973237, −8.606559891349605344038956058998, −7.22317289870293016838503490521, −6.49105109231236456136267837357, −4.65397508232973392412560533494, −2.77943473221641575026161701356, −2.00349366548623411071674463114,
1.13432239849526866074295216074, 3.44237075756310115590585017018, 4.93396000337896008576185696147, 6.11380662704711512290748065881, 7.51119500017597123912899865931, 7.960308303691833554698443469856, 9.381908713744275402906523587750, 9.932634166999336917723454699693, 10.73081365025886425095119712603, 11.85985239257556447826169469356