L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 7-s − 8-s + 9-s + 12-s − 4·13-s + 14-s + 16-s − 6·17-s − 18-s + 2·19-s − 21-s + 9·23-s − 24-s + 4·26-s + 27-s − 28-s + 6·29-s + 2·31-s − 32-s + 6·34-s + 36-s + 2·37-s − 2·38-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.288·12-s − 1.10·13-s + 0.267·14-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.458·19-s − 0.218·21-s + 1.87·23-s − 0.204·24-s + 0.784·26-s + 0.192·27-s − 0.188·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s + 1.02·34-s + 1/6·36-s + 0.328·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.432840965\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.432840965\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−7.999528815244640255090291034063, −7.43982372113404856619973073228, −6.73045719493607177983802045950, −6.26028737687743978694722727024, −4.94781456811313362686633867242, −4.55566372193790786638372149850, −3.22321592586749765746106671181, −2.76082470563987119013927356463, −1.85767085663082005073222422164, −0.65686582167224004510726526494,
0.65686582167224004510726526494, 1.85767085663082005073222422164, 2.76082470563987119013927356463, 3.22321592586749765746106671181, 4.55566372193790786638372149850, 4.94781456811313362686633867242, 6.26028737687743978694722727024, 6.73045719493607177983802045950, 7.43982372113404856619973073228, 7.999528815244640255090291034063