| L(s) = 1 | − 3-s − 2·9-s − 11-s + 13-s + 3·17-s + 4·19-s + 2·23-s + 5·27-s − 29-s + 6·31-s + 33-s + 2·37-s − 39-s + 10·41-s − 9·47-s − 3·51-s − 14·53-s − 4·57-s − 6·59-s + 4·61-s + 10·67-s − 2·69-s − 16·71-s − 10·73-s − 11·79-s + 81-s − 4·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s − 0.301·11-s + 0.277·13-s + 0.727·17-s + 0.917·19-s + 0.417·23-s + 0.962·27-s − 0.185·29-s + 1.07·31-s + 0.174·33-s + 0.328·37-s − 0.160·39-s + 1.56·41-s − 1.31·47-s − 0.420·51-s − 1.92·53-s − 0.529·57-s − 0.781·59-s + 0.512·61-s + 1.22·67-s − 0.240·69-s − 1.89·71-s − 1.17·73-s − 1.23·79-s + 1/9·81-s − 0.439·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.443212924\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.443212924\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 19 T + p T^{2} \) |
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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
−7.71429816416579041694272996589, −6.95797232804584945627039551824, −6.10700679427301114387348353623, −5.74785628831711034046654594763, −4.98272442874676693019513445909, −4.36461968543287943520088053262, −3.19092851830134267844902707259, −2.85361645033959248837859229323, −1.53783866800281636250261549941, −0.61178854200223729545576002674,
0.61178854200223729545576002674, 1.53783866800281636250261549941, 2.85361645033959248837859229323, 3.19092851830134267844902707259, 4.36461968543287943520088053262, 4.98272442874676693019513445909, 5.74785628831711034046654594763, 6.10700679427301114387348353623, 6.95797232804584945627039551824, 7.71429816416579041694272996589
