L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 2·7-s − 8-s + 9-s + 10-s − 12-s − 2·14-s + 15-s + 16-s − 17-s − 18-s − 2·19-s − 20-s − 2·21-s − 8·23-s + 24-s + 25-s − 27-s + 2·28-s + 6·29-s − 30-s + 4·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.755·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.534·14-s + 0.258·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.436·21-s − 1.66·23-s + 0.204·24-s + 1/5·25-s − 0.192·27-s + 0.377·28-s + 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 86190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.17009481272697, −13.70975781930179, −13.14764296120866, −12.35482193349957, −12.07369295102670, −11.66736972060902, −11.21096046525403, −10.62798132122817, −10.23070427662185, −9.839137948360787, −9.085691910368451, −8.526237612718664, −8.034790915985346, −7.836231586853556, −7.023121724798294, −6.459889919673686, −6.213304541470826, −5.269290204349956, −4.931768070075936, −4.198420511584625, −3.740738870121434, −2.852019322861986, −2.119866578489162, −1.570988768318831, −0.7565904053340946, 0,
0.7565904053340946, 1.570988768318831, 2.119866578489162, 2.852019322861986, 3.740738870121434, 4.198420511584625, 4.931768070075936, 5.269290204349956, 6.213304541470826, 6.459889919673686, 7.023121724798294, 7.836231586853556, 8.034790915985346, 8.526237612718664, 9.085691910368451, 9.839137948360787, 10.23070427662185, 10.62798132122817, 11.21096046525403, 11.66736972060902, 12.07369295102670, 12.35482193349957, 13.14764296120866, 13.70975781930179, 14.17009481272697