| L(s) = 1 | − 7-s + 11-s + 3·13-s + 2·17-s − 5·19-s + 4·23-s − 3·29-s − 6·31-s + 3·37-s + 8·43-s − 9·47-s + 49-s − 4·53-s + 9·59-s − 2·61-s + 5·67-s + 9·73-s − 77-s − 4·79-s − 2·83-s − 10·89-s − 3·91-s − 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.377·7-s + 0.301·11-s + 0.832·13-s + 0.485·17-s − 1.14·19-s + 0.834·23-s − 0.557·29-s − 1.07·31-s + 0.493·37-s + 1.21·43-s − 1.31·47-s + 1/7·49-s − 0.549·53-s + 1.17·59-s − 0.256·61-s + 0.610·67-s + 1.05·73-s − 0.113·77-s − 0.450·79-s − 0.219·83-s − 1.05·89-s − 0.314·91-s − 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.061033171\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.061033171\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−14.18565239086903, −13.63770880711046, −12.96503094059549, −12.78411316101788, −12.32691784487189, −11.46135715694759, −11.09590343569493, −10.82372156211901, −9.969631121243047, −9.673815498029960, −8.943418821175752, −8.662420642347431, −8.018265385596244, −7.409869074774715, −6.859535942588454, −6.321106944075893, −5.840673901087807, −5.264241767183408, −4.550111313928942, −3.891744721782130, −3.492351872424062, −2.765111180416812, −2.016230491010734, −1.319559618813250, −0.4962495068128487,
0.4962495068128487, 1.319559618813250, 2.016230491010734, 2.765111180416812, 3.492351872424062, 3.891744721782130, 4.550111313928942, 5.264241767183408, 5.840673901087807, 6.321106944075893, 6.859535942588454, 7.409869074774715, 8.018265385596244, 8.662420642347431, 8.943418821175752, 9.673815498029960, 9.969631121243047, 10.82372156211901, 11.09590343569493, 11.46135715694759, 12.32691784487189, 12.78411316101788, 12.96503094059549, 13.63770880711046, 14.18565239086903
