| L(s) = 1 | − 7·3-s − 7·7-s + 22·9-s + 9·11-s − 23·13-s − 41·17-s + 34·19-s + 49·21-s + 6·23-s + 35·27-s + 131·29-s + 4·31-s − 63·33-s − 26·37-s + 161·39-s − 260·41-s + 190·43-s − 167·47-s + 49·49-s + 287·51-s + 368·53-s − 238·57-s + 324·59-s − 164·61-s − 154·63-s − 200·67-s − 42·69-s + ⋯ |
| L(s) = 1 | − 1.34·3-s − 0.377·7-s + 0.814·9-s + 0.246·11-s − 0.490·13-s − 0.584·17-s + 0.410·19-s + 0.509·21-s + 0.0543·23-s + 0.249·27-s + 0.838·29-s + 0.0231·31-s − 0.332·33-s − 0.115·37-s + 0.661·39-s − 0.990·41-s + 0.673·43-s − 0.518·47-s + 1/7·49-s + 0.788·51-s + 0.953·53-s − 0.553·57-s + 0.714·59-s − 0.344·61-s − 0.307·63-s − 0.364·67-s − 0.0732·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 11 | \( 1 - 9 T + p^{3} T^{2} \) |
| 13 | \( 1 + 23 T + p^{3} T^{2} \) |
| 17 | \( 1 + 41 T + p^{3} T^{2} \) |
| 19 | \( 1 - 34 T + p^{3} T^{2} \) |
| 23 | \( 1 - 6 T + p^{3} T^{2} \) |
| 29 | \( 1 - 131 T + p^{3} T^{2} \) |
| 31 | \( 1 - 4 T + p^{3} T^{2} \) |
| 37 | \( 1 + 26 T + p^{3} T^{2} \) |
| 41 | \( 1 + 260 T + p^{3} T^{2} \) |
| 43 | \( 1 - 190 T + p^{3} T^{2} \) |
| 47 | \( 1 + 167 T + p^{3} T^{2} \) |
| 53 | \( 1 - 368 T + p^{3} T^{2} \) |
| 59 | \( 1 - 324 T + p^{3} T^{2} \) |
| 61 | \( 1 + 164 T + p^{3} T^{2} \) |
| 67 | \( 1 + 200 T + p^{3} T^{2} \) |
| 71 | \( 1 - 784 T + p^{3} T^{2} \) |
| 73 | \( 1 - 410 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1211 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1132 T + p^{3} T^{2} \) |
| 89 | \( 1 + 72 T + p^{3} T^{2} \) |
| 97 | \( 1 - 707 T + p^{3} T^{2} \) |
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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
−8.908250920768921909450233400665, −7.87639271882247902089173789298, −6.80270392681718424509064028916, −6.42715273720300435512060635282, −5.40301280995019038722878177713, −4.81496496483145778253311920920, −3.72980914095461117555838914175, −2.45860489337839172791612642862, −1.01917131749344265039712905001, 0,
1.01917131749344265039712905001, 2.45860489337839172791612642862, 3.72980914095461117555838914175, 4.81496496483145778253311920920, 5.40301280995019038722878177713, 6.42715273720300435512060635282, 6.80270392681718424509064028916, 7.87639271882247902089173789298, 8.908250920768921909450233400665
