L(s) = 1 | − 27·9-s + 92·13-s + 104·17-s − 130·29-s − 396·37-s + 230·41-s − 343·49-s − 572·53-s + 830·61-s + 592·73-s + 729·81-s + 1.67e3·89-s + 1.81e3·97-s − 598·101-s + 1.74e3·109-s + 1.32e3·113-s − 2.48e3·117-s + ⋯ |
L(s) = 1 | − 9-s + 1.96·13-s + 1.48·17-s − 0.832·29-s − 1.75·37-s + 0.876·41-s − 49-s − 1.48·53-s + 1.74·61-s + 0.949·73-s + 81-s + 1.98·89-s + 1.90·97-s − 0.589·101-s + 1.53·109-s + 1.10·113-s − 1.96·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.170747332\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.170747332\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 92 T + p^{3} T^{2} \) |
| 17 | \( 1 - 104 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + 396 T + p^{3} T^{2} \) |
| 41 | \( 1 - 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 + 572 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 830 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 592 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 - 1670 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1816 T + p^{3} T^{2} \) |
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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
−8.910638815477927300418543725229, −8.322731025003355346338596659987, −7.60334988348691863282311097380, −6.43705211097272875357808887672, −5.82621934776736941704139075647, −5.09289003427586174508546352719, −3.67128040244791292349057637806, −3.26210713818336384030711292775, −1.82182526404571793481162085201, −0.72776802490996063526525155967,
0.72776802490996063526525155967, 1.82182526404571793481162085201, 3.26210713818336384030711292775, 3.67128040244791292349057637806, 5.09289003427586174508546352719, 5.82621934776736941704139075647, 6.43705211097272875357808887672, 7.60334988348691863282311097380, 8.322731025003355346338596659987, 8.910638815477927300418543725229