L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s + 2·13-s + 16-s + 2·17-s + 18-s + 2·19-s − 8·23-s − 24-s + 2·26-s − 27-s − 8·29-s + 4·31-s + 32-s + 2·34-s + 36-s + 6·37-s + 2·38-s − 2·39-s + 10·41-s − 2·43-s − 8·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.288·12-s + 0.554·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 0.458·19-s − 1.66·23-s − 0.204·24-s + 0.392·26-s − 0.192·27-s − 1.48·29-s + 0.718·31-s + 0.176·32-s + 0.342·34-s + 1/6·36-s + 0.986·37-s + 0.324·38-s − 0.320·39-s + 1.56·41-s − 0.304·43-s − 1.17·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.729303724\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.729303724\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−7.84745293504310182206937228375, −7.05731450515241457980992209453, −6.33385015448198061798453543002, −5.69407662874022337594626370706, −5.30083129047836043881575068002, −4.13443133588443750202461924520, −3.89844472964905616102389104102, −2.77072751197698490599085887569, −1.86120969838198474002846896926, −0.78106727636813346319047125036,
0.78106727636813346319047125036, 1.86120969838198474002846896926, 2.77072751197698490599085887569, 3.89844472964905616102389104102, 4.13443133588443750202461924520, 5.30083129047836043881575068002, 5.69407662874022337594626370706, 6.33385015448198061798453543002, 7.05731450515241457980992209453, 7.84745293504310182206937228375