L(s) = 1 | + i·2-s + 3-s − 4-s − i·5-s + i·6-s + 0.643i·7-s − i·8-s + 9-s + 10-s + 4.40i·11-s − 12-s − 0.643·14-s − i·15-s + 16-s − 0.939·17-s + i·18-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 0.447i·5-s + 0.408i·6-s + 0.243i·7-s − 0.353i·8-s + 0.333·9-s + 0.316·10-s + 1.32i·11-s − 0.288·12-s − 0.171·14-s − 0.258i·15-s + 0.250·16-s − 0.227·17-s + 0.235i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5070 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.759578350\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.759578350\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + iT \) |
| 13 | \( 1 \) |
good | 7 | \( 1 - 0.643iT - 7T^{2} \) |
| 11 | \( 1 - 4.40iT - 11T^{2} \) |
| 17 | \( 1 + 0.939T + 17T^{2} \) |
| 19 | \( 1 + 2.75iT - 19T^{2} \) |
| 23 | \( 1 - 2.04T + 23T^{2} \) |
| 29 | \( 1 + 0.149T + 29T^{2} \) |
| 31 | \( 1 + 2.08iT - 31T^{2} \) |
| 37 | \( 1 - 6.07iT - 37T^{2} \) |
| 41 | \( 1 - 5.74iT - 41T^{2} \) |
| 43 | \( 1 + 4.69T + 43T^{2} \) |
| 47 | \( 1 - 7iT - 47T^{2} \) |
| 53 | \( 1 - 1.68T + 53T^{2} \) |
| 59 | \( 1 - 7.85iT - 59T^{2} \) |
| 61 | \( 1 + 12.0T + 61T^{2} \) |
| 67 | \( 1 + 1.45iT - 67T^{2} \) |
| 71 | \( 1 - 7.03iT - 71T^{2} \) |
| 73 | \( 1 - 14.8iT - 73T^{2} \) |
| 79 | \( 1 - 0.929T + 79T^{2} \) |
| 83 | \( 1 + 11.2iT - 83T^{2} \) |
| 89 | \( 1 + 7.47iT - 89T^{2} \) |
| 97 | \( 1 - 5.05iT - 97T^{2} \) |
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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
−8.527732083008923415566087467315, −7.69255959017892427861800085908, −7.18815131371889765224929917862, −6.47257167863705888701548148717, −5.60514136865861696400643970732, −4.62893893186490543550712198587, −4.43785907864608096397679481531, −3.20418376365781325115226018298, −2.28313307707080526488448093929, −1.21260996413552044409472272698,
0.45164107774321531685043783503, 1.66638891245132606962745416442, 2.58572187856610350622794271558, 3.44973874229984704262759082524, 3.80762054526566905777518459465, 4.91404753015908897957852061160, 5.75140228574212187561810992811, 6.53430779062340565228653116284, 7.41105481113175808164358406259, 8.077143688341806538437001525105