L(s) = 1 | − 2-s + 4-s − 7-s − 8-s + 3·11-s − 5·13-s + 14-s + 16-s − 7·19-s − 3·22-s − 6·23-s + 5·26-s − 28-s − 6·29-s − 4·31-s − 32-s − 2·37-s + 7·38-s − 3·41-s + 43-s + 3·44-s + 6·46-s − 3·47-s + 49-s − 5·52-s − 9·53-s + 56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s + 0.904·11-s − 1.38·13-s + 0.267·14-s + 1/4·16-s − 1.60·19-s − 0.639·22-s − 1.25·23-s + 0.980·26-s − 0.188·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.328·37-s + 1.13·38-s − 0.468·41-s + 0.152·43-s + 0.452·44-s + 0.884·46-s − 0.437·47-s + 1/7·49-s − 0.693·52-s − 1.23·53-s + 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6830563848\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6830563848\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 3 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.79980340132329946748917572315, −6.95630256007367246800491223129, −6.57040873040844082848576651742, −5.82201734940959471254317323776, −4.97794781017817303425016676386, −4.06891154825711709658222185843, −3.46531667791355369480874825879, −2.25204495118066426625760749971, −1.88034482244788806821330679674, −0.41706266611791957510417112120,
0.41706266611791957510417112120, 1.88034482244788806821330679674, 2.25204495118066426625760749971, 3.46531667791355369480874825879, 4.06891154825711709658222185843, 4.97794781017817303425016676386, 5.82201734940959471254317323776, 6.57040873040844082848576651742, 6.95630256007367246800491223129, 7.79980340132329946748917572315