| L(s) = 1 | − 5-s + 7-s − 3·9-s + 13-s − 2·17-s − 8·19-s − 4·25-s + 9·29-s − 4·31-s − 35-s + 6·37-s − 5·41-s − 8·43-s + 3·45-s + 49-s − 9·53-s − 5·61-s − 3·63-s − 65-s − 16·67-s + 12·71-s − 73-s − 4·79-s + 9·81-s − 12·83-s + 2·85-s + 9·89-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s − 9-s + 0.277·13-s − 0.485·17-s − 1.83·19-s − 4/5·25-s + 1.67·29-s − 0.718·31-s − 0.169·35-s + 0.986·37-s − 0.780·41-s − 1.21·43-s + 0.447·45-s + 1/7·49-s − 1.23·53-s − 0.640·61-s − 0.377·63-s − 0.124·65-s − 1.95·67-s + 1.42·71-s − 0.117·73-s − 0.450·79-s + 81-s − 1.31·83-s + 0.216·85-s + 0.953·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - T + p T^{2} \) |
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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
−13.11603905502589, −12.69608607368191, −12.09448026628069, −11.69497506003868, −11.33995971244099, −10.86887901948019, −10.42362453406268, −10.02565710496589, −9.190534037434870, −8.891994360272644, −8.386449911204058, −8.036956964250452, −7.681445761999329, −6.786569965713322, −6.492918690931907, −6.035702617345927, −5.480065073816194, −4.781674996877597, −4.432880135385412, −3.935952004944612, −3.188308423165901, −2.814971829516558, −2.019778964959231, −1.649714884664109, −0.5932125666133799, 0,
0.5932125666133799, 1.649714884664109, 2.019778964959231, 2.814971829516558, 3.188308423165901, 3.935952004944612, 4.432880135385412, 4.781674996877597, 5.480065073816194, 6.035702617345927, 6.492918690931907, 6.786569965713322, 7.681445761999329, 8.036956964250452, 8.386449911204058, 8.891994360272644, 9.190534037434870, 10.02565710496589, 10.42362453406268, 10.86887901948019, 11.33995971244099, 11.69497506003868, 12.09448026628069, 12.69608607368191, 13.11603905502589
