| L(s) = 1 | − 2-s + 4-s + 3.35·7-s − 8-s + 1.61·11-s − 1.35·13-s − 3.35·14-s + 16-s − 6.96·17-s − 19-s − 1.61·22-s + 1.35·23-s + 1.35·26-s + 3.35·28-s − 3.61·29-s − 2.31·31-s − 32-s + 6.96·34-s + 11.2·37-s + 38-s + 3.35·41-s − 10.3·43-s + 1.61·44-s − 1.35·46-s + 4.57·47-s + 4.22·49-s − 1.35·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.26·7-s − 0.353·8-s + 0.486·11-s − 0.374·13-s − 0.895·14-s + 0.250·16-s − 1.68·17-s − 0.229·19-s − 0.343·22-s + 0.281·23-s + 0.264·26-s + 0.633·28-s − 0.670·29-s − 0.415·31-s − 0.176·32-s + 1.19·34-s + 1.85·37-s + 0.162·38-s + 0.523·41-s − 1.57·43-s + 0.243·44-s − 0.199·46-s + 0.667·47-s + 0.603·49-s − 0.187·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 - 3.35T + 7T^{2} \) |
| 11 | \( 1 - 1.61T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 + 6.96T + 17T^{2} \) |
| 23 | \( 1 - 1.35T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 2.31T + 31T^{2} \) |
| 37 | \( 1 - 11.2T + 37T^{2} \) |
| 41 | \( 1 - 3.35T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 47 | \( 1 - 4.57T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 1.03T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 9.92T + 67T^{2} \) |
| 71 | \( 1 - 0.775T + 71T^{2} \) |
| 73 | \( 1 + 3.22T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 2.57T + 89T^{2} \) |
| 97 | \( 1 + 1.16T + 97T^{2} \) |
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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
−7.66766062346074063599026868216, −6.80112782038922583156676780769, −6.28695242462468738448390702846, −5.33155595652291036965448604762, −4.59799295266907272373988751705, −4.02136733261253696001527034117, −2.79246547229151501379194570727, −2.01682000908055418583122834502, −1.31568274689221893632334716861, 0,
1.31568274689221893632334716861, 2.01682000908055418583122834502, 2.79246547229151501379194570727, 4.02136733261253696001527034117, 4.59799295266907272373988751705, 5.33155595652291036965448604762, 6.28695242462468738448390702846, 6.80112782038922583156676780769, 7.66766062346074063599026868216
