| 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
| \(L(2)\) |
\(\approx\) |
\(0.9707877197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9707877197\) |
| \(L(\frac{5}{2})\) |
|
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
−9.153414178536364170474972581429, −8.236887585124414802485640997880, −7.45721832291116743968145268197, −6.67893136648116745588846640316, −5.73172590299788706605911013151, −4.92189185495959417070482783843, −4.09058620606015797332101007998, −2.73650619566920242331990061996, −2.20328927761097337870816619045, −0.44360603596238490082235246168,
0.44360603596238490082235246168, 2.20328927761097337870816619045, 2.73650619566920242331990061996, 4.09058620606015797332101007998, 4.92189185495959417070482783843, 5.73172590299788706605911013151, 6.67893136648116745588846640316, 7.45721832291116743968145268197, 8.236887585124414802485640997880, 9.153414178536364170474972581429
