| L(s) = 1 | + 201.·3-s − 2.30e3·7-s + 2.09e4·9-s + 2.17e4·11-s − 9.51e4·13-s + 1.50e4·17-s − 1.12e6·19-s − 4.63e5·21-s + 2.00e6·23-s + 2.53e5·27-s − 1.18e6·29-s − 1.92e6·31-s + 4.39e6·33-s − 4.44e6·37-s − 1.91e7·39-s − 2.19e7·41-s + 2.53e7·43-s − 3.64e7·47-s − 3.50e7·49-s + 3.03e6·51-s − 4.54e7·53-s − 2.25e8·57-s − 1.32e8·59-s + 1.55e8·61-s − 4.81e7·63-s + 6.79e6·67-s + 4.04e8·69-s + ⋯ |
| L(s) = 1 | + 1.43·3-s − 0.362·7-s + 1.06·9-s + 0.448·11-s − 0.924·13-s + 0.0437·17-s − 1.97·19-s − 0.520·21-s + 1.49·23-s + 0.0917·27-s − 0.311·29-s − 0.374·31-s + 0.644·33-s − 0.390·37-s − 1.32·39-s − 1.21·41-s + 1.12·43-s − 1.09·47-s − 0.868·49-s + 0.0628·51-s − 0.791·53-s − 2.83·57-s − 1.42·59-s + 1.43·61-s − 0.385·63-s + 0.0412·67-s + 2.15·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 201.T + 1.96e4T^{2} \) |
| 7 | \( 1 + 2.30e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 2.17e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 9.51e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 1.50e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.12e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.00e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.18e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.92e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 4.44e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.19e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 2.53e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.64e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 4.54e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.32e8T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.55e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 6.79e6T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.35e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.76e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.60e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.42e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.25e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.55e9T + 7.60e17T^{2} \) |
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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
−10.04823931919720318079165695933, −9.154157933302894606362217213844, −8.497372206342871181984239360061, −7.41419427227386726036572546589, −6.45975384500906083239148904636, −4.80626354832094754119728930835, −3.65503809324823083243895879078, −2.69229278942826261375275945360, −1.71451427309600999975436111224, 0,
1.71451427309600999975436111224, 2.69229278942826261375275945360, 3.65503809324823083243895879078, 4.80626354832094754119728930835, 6.45975384500906083239148904636, 7.41419427227386726036572546589, 8.497372206342871181984239360061, 9.154157933302894606362217213844, 10.04823931919720318079165695933
