| L(s) = 1 | + 5-s − 3·11-s + 13-s − 3·17-s + 23-s + 25-s − 6·29-s − 10·31-s − 9·37-s − 3·41-s − 10·43-s − 2·47-s − 10·53-s − 3·55-s + 9·59-s − 2·61-s + 65-s − 5·67-s − 7·73-s − 79-s − 10·83-s − 3·85-s + 3·89-s + 7·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.208·23-s + 1/5·25-s − 1.11·29-s − 1.79·31-s − 1.47·37-s − 0.468·41-s − 1.52·43-s − 0.291·47-s − 1.37·53-s − 0.404·55-s + 1.17·59-s − 0.256·61-s + 0.124·65-s − 0.610·67-s − 0.819·73-s − 0.112·79-s − 1.09·83-s − 0.325·85-s + 0.317·89-s + 0.710·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−13.29512542452763, −13.01560744749755, −12.63777362456245, −12.01213489768863, −11.42749011034710, −11.03705360175882, −10.68727997620229, −10.06231461916285, −9.811151157315575, −9.052829455795354, −8.791105617313412, −8.330349179856942, −7.629680656566302, −7.265952379576689, −6.762486773148918, −6.219895441651306, −5.649865316294650, −5.173596895938983, −4.883085239240552, −4.082557728924981, −3.487176089109689, −3.103813446107409, −2.291320886568751, −1.830021634585894, −1.343296600604423, 0, 0,
1.343296600604423, 1.830021634585894, 2.291320886568751, 3.103813446107409, 3.487176089109689, 4.082557728924981, 4.883085239240552, 5.173596895938983, 5.649865316294650, 6.219895441651306, 6.762486773148918, 7.265952379576689, 7.629680656566302, 8.330349179856942, 8.791105617313412, 9.052829455795354, 9.811151157315575, 10.06231461916285, 10.68727997620229, 11.03705360175882, 11.42749011034710, 12.01213489768863, 12.63777362456245, 13.01560744749755, 13.29512542452763
