L(s) = 1 | + 1.77·2-s + 3-s + 1.16·4-s − 0.681·5-s + 1.77·6-s − 7-s − 1.48·8-s + 9-s − 1.21·10-s − 0.590·11-s + 1.16·12-s − 1.77·14-s − 0.681·15-s − 4.97·16-s − 6.29·17-s + 1.77·18-s + 3.58·19-s − 0.794·20-s − 21-s − 1.05·22-s + 3.68·23-s − 1.48·24-s − 4.53·25-s + 27-s − 1.16·28-s − 6.45·29-s − 1.21·30-s + ⋯ |
L(s) = 1 | + 1.25·2-s + 0.577·3-s + 0.582·4-s − 0.304·5-s + 0.726·6-s − 0.377·7-s − 0.524·8-s + 0.333·9-s − 0.383·10-s − 0.177·11-s + 0.336·12-s − 0.475·14-s − 0.176·15-s − 1.24·16-s − 1.52·17-s + 0.419·18-s + 0.822·19-s − 0.177·20-s − 0.218·21-s − 0.223·22-s + 0.768·23-s − 0.302·24-s − 0.907·25-s + 0.192·27-s − 0.220·28-s − 1.19·29-s − 0.221·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 - 1.77T + 2T^{2} \) |
| 5 | \( 1 + 0.681T + 5T^{2} \) |
| 11 | \( 1 + 0.590T + 11T^{2} \) |
| 17 | \( 1 + 6.29T + 17T^{2} \) |
| 19 | \( 1 - 3.58T + 19T^{2} \) |
| 23 | \( 1 - 3.68T + 23T^{2} \) |
| 29 | \( 1 + 6.45T + 29T^{2} \) |
| 31 | \( 1 + 4.30T + 31T^{2} \) |
| 37 | \( 1 + 10.5T + 37T^{2} \) |
| 41 | \( 1 + 0.157T + 41T^{2} \) |
| 43 | \( 1 - 4.72T + 43T^{2} \) |
| 47 | \( 1 + 11.2T + 47T^{2} \) |
| 53 | \( 1 - 12.2T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 + 7.48T + 67T^{2} \) |
| 71 | \( 1 - 6.99T + 71T^{2} \) |
| 73 | \( 1 + 7.30T + 73T^{2} \) |
| 79 | \( 1 + 3.33T + 79T^{2} \) |
| 83 | \( 1 - 13.6T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 + 10.6T + 97T^{2} \) |
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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
−8.181057736224708066738105087458, −7.17089937409161714974602859865, −6.75662388098317401752712309733, −5.68687111755776709727537232166, −5.11666860668871355048409296453, −4.15505694261639731417208925094, −3.63828537232105531652465196026, −2.84377383581018158888410372273, −1.91452614718409322420414540466, 0,
1.91452614718409322420414540466, 2.84377383581018158888410372273, 3.63828537232105531652465196026, 4.15505694261639731417208925094, 5.11666860668871355048409296453, 5.68687111755776709727537232166, 6.75662388098317401752712309733, 7.17089937409161714974602859865, 8.181057736224708066738105087458