| L(s) = 1 | + 222.·3-s + 7.88e3·7-s + 2.96e4·9-s − 3.95e4·11-s − 1.77e5·13-s − 1.65e5·17-s − 9.88e4·19-s + 1.75e6·21-s + 4.70e5·23-s + 2.21e6·27-s − 4.65e6·29-s + 8.86e6·31-s − 8.78e6·33-s − 9.69e6·37-s − 3.94e7·39-s − 3.27e7·41-s + 1.38e7·43-s − 2.02e7·47-s + 2.18e7·49-s − 3.66e7·51-s − 9.75e7·53-s − 2.19e7·57-s − 7.95e7·59-s − 1.54e8·61-s + 2.34e8·63-s + 3.11e8·67-s + 1.04e8·69-s + ⋯ |
| L(s) = 1 | + 1.58·3-s + 1.24·7-s + 1.50·9-s − 0.814·11-s − 1.72·13-s − 0.479·17-s − 0.174·19-s + 1.96·21-s + 0.350·23-s + 0.803·27-s − 1.22·29-s + 1.72·31-s − 1.28·33-s − 0.850·37-s − 2.73·39-s − 1.80·41-s + 0.615·43-s − 0.604·47-s + 0.541·49-s − 0.759·51-s − 1.69·53-s − 0.275·57-s − 0.854·59-s − 1.42·61-s + 1.87·63-s + 1.88·67-s + 0.554·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 222.T + 1.96e4T^{2} \) |
| 7 | \( 1 - 7.88e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.95e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.77e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.65e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 9.88e4T + 3.22e11T^{2} \) |
| 23 | \( 1 - 4.70e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.65e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.86e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 9.69e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 3.27e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.38e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.02e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.75e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 7.95e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.54e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.11e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 9.08e6T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.83e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.58e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 3.32e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.41e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 3.69e8T + 7.60e17T^{2} \) |
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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
−9.219688786657865715750411635758, −8.174489356651595187095607655728, −7.84680838984724409387135602327, −6.89767977503330204853556746518, −5.12501772000331242202128295787, −4.52414267690357665568073970323, −3.18384564210998022593087036811, −2.33139499245242815807336157320, −1.65598778980701564074287778846, 0,
1.65598778980701564074287778846, 2.33139499245242815807336157320, 3.18384564210998022593087036811, 4.52414267690357665568073970323, 5.12501772000331242202128295787, 6.89767977503330204853556746518, 7.84680838984724409387135602327, 8.174489356651595187095607655728, 9.219688786657865715750411635758
