| L(s) = 1 | − 2·2-s + 4·4-s − 0.803·5-s − 2.39·7-s − 8·8-s + 1.60·10-s − 59.5·11-s + 77.7·13-s + 4.78·14-s + 16·16-s − 2.84·17-s − 118.·19-s − 3.21·20-s + 119.·22-s − 110.·23-s − 124.·25-s − 155.·26-s − 9.56·28-s − 125.·29-s + 63.0·31-s − 32·32-s + 5.68·34-s + 1.92·35-s − 227.·37-s + 236.·38-s + 6.43·40-s − 324.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.0718·5-s − 0.129·7-s − 0.353·8-s + 0.0508·10-s − 1.63·11-s + 1.65·13-s + 0.0913·14-s + 0.250·16-s − 0.0405·17-s − 1.42·19-s − 0.0359·20-s + 1.15·22-s − 1.00·23-s − 0.994·25-s − 1.17·26-s − 0.0645·28-s − 0.804·29-s + 0.365·31-s − 0.176·32-s + 0.0286·34-s + 0.00928·35-s − 1.01·37-s + 1.01·38-s + 0.0254·40-s − 1.23·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 0.803T + 125T^{2} \) |
| 7 | \( 1 + 2.39T + 343T^{2} \) |
| 11 | \( 1 + 59.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 77.7T + 2.19e3T^{2} \) |
| 17 | \( 1 + 2.84T + 4.91e3T^{2} \) |
| 19 | \( 1 + 118.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 110.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 125.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 63.0T + 2.97e4T^{2} \) |
| 37 | \( 1 + 227.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 324.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 272.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 4.93T + 1.03e5T^{2} \) |
| 53 | \( 1 - 598.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 670.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 464.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 769.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 611.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 923.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 39.8T + 4.93e5T^{2} \) |
| 83 | \( 1 + 443.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 78.4T + 7.04e5T^{2} \) |
| 97 | \( 1 + 91.0T + 9.12e5T^{2} \) |
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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
−11.67755049003422031342868460763, −10.69254113223151628980573646813, −10.03914823785766153218731762723, −8.568304504795535210966827679765, −8.043930522295126479344748887877, −6.62877909578204508362771540617, −5.52454323165698680593592555200, −3.71943993641938030771600832543, −2.04871865276272112224892157250, 0,
2.04871865276272112224892157250, 3.71943993641938030771600832543, 5.52454323165698680593592555200, 6.62877909578204508362771540617, 8.043930522295126479344748887877, 8.568304504795535210966827679765, 10.03914823785766153218731762723, 10.69254113223151628980573646813, 11.67755049003422031342868460763
