| L(s) = 1 | − 3·3-s − 5·7-s + 6·9-s + 4·11-s + 13-s + 3·17-s − 19-s + 15·21-s + 7·23-s − 9·27-s − 3·29-s + 2·31-s − 12·33-s + 2·37-s − 3·39-s − 6·41-s + 6·43-s + 18·49-s − 9·51-s + 13·53-s + 3·57-s + 9·59-s − 12·61-s − 30·63-s − 3·67-s − 21·69-s − 11·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.88·7-s + 2·9-s + 1.20·11-s + 0.277·13-s + 0.727·17-s − 0.229·19-s + 3.27·21-s + 1.45·23-s − 1.73·27-s − 0.557·29-s + 0.359·31-s − 2.08·33-s + 0.328·37-s − 0.480·39-s − 0.937·41-s + 0.914·43-s + 18/7·49-s − 1.26·51-s + 1.78·53-s + 0.397·57-s + 1.17·59-s − 1.53·61-s − 3.77·63-s − 0.366·67-s − 2.52·69-s − 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8087246128\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8087246128\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−7.44716628247100608220869942622, −6.87686093893102620296125428620, −6.46146665314716960456556084861, −5.87264009679714677145341907258, −5.34814404867396916909799829210, −4.30956041572834653164515437556, −3.70582580501801640360877606053, −2.85255620330002162521681921778, −1.31795279825686175873821504061, −0.55393834765254454631302819206,
0.55393834765254454631302819206, 1.31795279825686175873821504061, 2.85255620330002162521681921778, 3.70582580501801640360877606053, 4.30956041572834653164515437556, 5.34814404867396916909799829210, 5.87264009679714677145341907258, 6.46146665314716960456556084861, 6.87686093893102620296125428620, 7.44716628247100608220869942622
