L(s) = 1 | − 10·13-s − 30·17-s − 42·29-s + 70·37-s − 18·41-s + 49·49-s + 90·53-s − 22·61-s + 110·73-s + 78·89-s − 130·97-s + 198·101-s − 182·109-s − 30·113-s + ⋯ |
L(s) = 1 | − 0.769·13-s − 1.76·17-s − 1.44·29-s + 1.89·37-s − 0.439·41-s + 49-s + 1.69·53-s − 0.360·61-s + 1.50·73-s + 0.876·89-s − 1.34·97-s + 1.96·101-s − 1.66·109-s − 0.265·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.477078123\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.477078123\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 10 T + p^{2} T^{2} \) |
| 17 | \( 1 + 30 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 + 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 - 70 T + p^{2} T^{2} \) |
| 41 | \( 1 + 18 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 - 90 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 110 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 78 T + p^{2} T^{2} \) |
| 97 | \( 1 + 130 T + p^{2} T^{2} \) |
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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
−8.415142136187616579508631786662, −7.56094630004435378588468174713, −6.96499306892419703342936465859, −6.17825050538555368412861176232, −5.35217024478186232096209907408, −4.50577717994355236600796349677, −3.84896528356747111042980790382, −2.63188970949711711240577990984, −1.98923442557520889449285097685, −0.55088988637239689374962881884,
0.55088988637239689374962881884, 1.98923442557520889449285097685, 2.63188970949711711240577990984, 3.84896528356747111042980790382, 4.50577717994355236600796349677, 5.35217024478186232096209907408, 6.17825050538555368412861176232, 6.96499306892419703342936465859, 7.56094630004435378588468174713, 8.415142136187616579508631786662