| L(s) = 1 | − 3·3-s + 6·9-s − 3·11-s − 13-s − 5·17-s − 8·19-s − 2·23-s − 9·27-s + 29-s + 2·31-s + 9·33-s + 10·37-s + 3·39-s + 6·41-s − 4·43-s + 11·47-s + 15·51-s + 6·53-s + 24·57-s − 10·59-s − 10·67-s + 6·69-s − 10·73-s − 7·79-s + 9·81-s − 12·83-s − 3·87-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 2·9-s − 0.904·11-s − 0.277·13-s − 1.21·17-s − 1.83·19-s − 0.417·23-s − 1.73·27-s + 0.185·29-s + 0.359·31-s + 1.56·33-s + 1.64·37-s + 0.480·39-s + 0.937·41-s − 0.609·43-s + 1.60·47-s + 2.10·51-s + 0.824·53-s + 3.17·57-s − 1.30·59-s − 1.22·67-s + 0.722·69-s − 1.17·73-s − 0.787·79-s + 81-s − 1.31·83-s − 0.321·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 7 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 3 T + p T^{2} \) |
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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
−14.26901211944386, −13.51267142945659, −13.08157445606856, −12.73977991046492, −12.27762738739661, −11.70516662600861, −11.25295462476238, −10.72890870517175, −10.51640407528730, −10.00427258825099, −9.330444371288163, −8.679261817136362, −8.148340474589096, −7.417470285761110, −7.005445232929923, −6.386965428051481, −5.886706797050033, −5.679779840634059, −4.717778714350850, −4.416106580806908, −4.124574695776233, −2.823899734976649, −2.326851571195710, −1.544834168982458, −0.5769087524902653, 0,
0.5769087524902653, 1.544834168982458, 2.326851571195710, 2.823899734976649, 4.124574695776233, 4.416106580806908, 4.717778714350850, 5.679779840634059, 5.886706797050033, 6.386965428051481, 7.005445232929923, 7.417470285761110, 8.148340474589096, 8.679261817136362, 9.330444371288163, 10.00427258825099, 10.51640407528730, 10.72890870517175, 11.25295462476238, 11.70516662600861, 12.27762738739661, 12.73977991046492, 13.08157445606856, 13.51267142945659, 14.26901211944386
