| L(s) = 1 | + 9·9-s + 24·13-s − 16·17-s + 42·29-s − 24·37-s − 18·41-s + 49·49-s − 56·53-s − 22·61-s + 96·73-s + 81·81-s + 78·89-s + 144·97-s + 198·101-s + 182·109-s − 224·113-s + 216·117-s + ⋯ |
| L(s) = 1 | + 9-s + 1.84·13-s − 0.941·17-s + 1.44·29-s − 0.648·37-s − 0.439·41-s + 49-s − 1.05·53-s − 0.360·61-s + 1.31·73-s + 81-s + 0.876·89-s + 1.48·97-s + 1.96·101-s + 1.66·109-s − 1.98·113-s + 1.84·117-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.397458864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.397458864\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( ( 1 - p T )( 1 + p T ) \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 - 24 T + p^{2} T^{2} \) |
| 17 | \( 1 + 16 T + p^{2} T^{2} \) |
| 19 | \( ( 1 - p T )( 1 + p T ) \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( 1 - 42 T + p^{2} T^{2} \) |
| 31 | \( ( 1 - p T )( 1 + p T ) \) |
| 37 | \( 1 + 24 T + p^{2} T^{2} \) |
| 41 | \( 1 + 18 T + p^{2} T^{2} \) |
| 43 | \( ( 1 - p T )( 1 + p T ) \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( 1 + 56 T + p^{2} T^{2} \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 22 T + p^{2} T^{2} \) |
| 67 | \( ( 1 - p T )( 1 + p T ) \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 - 96 T + p^{2} T^{2} \) |
| 79 | \( ( 1 - p T )( 1 + p T ) \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( 1 - 78 T + p^{2} T^{2} \) |
| 97 | \( 1 - 144 T + p^{2} T^{2} \) |
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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
−9.067451523933460378813938746497, −8.548334913378864213531731454821, −7.64931462495824313503923701824, −6.65140066064621457558447216476, −6.20966855530344367496724951090, −4.97974795175139794773435964667, −4.15513060907452172925356557342, −3.32582005323866187656244806748, −1.95153783801062952979524762458, −0.919366115397596420390159720012,
0.919366115397596420390159720012, 1.95153783801062952979524762458, 3.32582005323866187656244806748, 4.15513060907452172925356557342, 4.97974795175139794773435964667, 6.20966855530344367496724951090, 6.65140066064621457558447216476, 7.64931462495824313503923701824, 8.548334913378864213531731454821, 9.067451523933460378813938746497
