L(s) = 1 | + 2-s + 4-s + 5-s + 8-s + 10-s − 13-s + 16-s + 6·17-s + 20-s + 4·23-s + 25-s − 26-s + 10·29-s + 32-s + 6·34-s − 6·37-s + 40-s − 2·41-s − 4·43-s + 4·46-s − 7·49-s + 50-s − 52-s + 6·53-s + 10·58-s + 6·61-s + 64-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.353·8-s + 0.316·10-s − 0.277·13-s + 1/4·16-s + 1.45·17-s + 0.223·20-s + 0.834·23-s + 1/5·25-s − 0.196·26-s + 1.85·29-s + 0.176·32-s + 1.02·34-s − 0.986·37-s + 0.158·40-s − 0.312·41-s − 0.609·43-s + 0.589·46-s − 49-s + 0.141·50-s − 0.138·52-s + 0.824·53-s + 1.31·58-s + 0.768·61-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1170 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.880913927\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.880913927\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−10.07123462831113686449175462085, −8.955383165969649286288406642539, −8.062771471643955273434189645297, −7.11791279895358735884585299226, −6.36199631841087100215380294304, −5.39759482678390501023193939738, −4.78086243570270981436649484362, −3.52270970074791721181391566206, −2.67714551530238471219197858825, −1.30131976065987787395283637597,
1.30131976065987787395283637597, 2.67714551530238471219197858825, 3.52270970074791721181391566206, 4.78086243570270981436649484362, 5.39759482678390501023193939738, 6.36199631841087100215380294304, 7.11791279895358735884585299226, 8.062771471643955273434189645297, 8.955383165969649286288406642539, 10.07123462831113686449175462085