| L(s) = 1 | − 3-s + 3·7-s + 9-s − 2·11-s + 2·13-s + 7·17-s − 6·19-s − 3·21-s − 23-s − 27-s − 9·29-s + 9·31-s + 2·33-s + 7·37-s − 2·39-s + 5·41-s − 8·47-s + 2·49-s − 7·51-s + 11·53-s + 6·57-s + 9·59-s + 3·63-s + 3·67-s + 69-s + 3·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.13·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s + 1.69·17-s − 1.37·19-s − 0.654·21-s − 0.208·23-s − 0.192·27-s − 1.67·29-s + 1.61·31-s + 0.348·33-s + 1.15·37-s − 0.320·39-s + 0.780·41-s − 1.16·47-s + 2/7·49-s − 0.980·51-s + 1.51·53-s + 0.794·57-s + 1.17·59-s + 0.377·63-s + 0.366·67-s + 0.120·69-s + 0.356·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.918586391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918586391\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 3 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.069669155801640894146017525841, −7.36913090045138333376970374213, −6.44660273835631511784393340976, −5.73674719055418870802783672106, −5.24541142164515214520039056454, −4.41742680466752310626581711630, −3.77422385356999700348037806420, −2.61537486554548359682980367336, −1.69346935016057027327943982766, −0.76122223760427925475297547796,
0.76122223760427925475297547796, 1.69346935016057027327943982766, 2.61537486554548359682980367336, 3.77422385356999700348037806420, 4.41742680466752310626581711630, 5.24541142164515214520039056454, 5.73674719055418870802783672106, 6.44660273835631511784393340976, 7.36913090045138333376970374213, 8.069669155801640894146017525841
