L(s) = 1 | + 3-s − 7-s + 9-s + 6·11-s + 5·13-s − 6·17-s + 5·19-s − 21-s − 6·23-s + 27-s − 6·29-s − 31-s + 6·33-s + 2·37-s + 5·39-s − 43-s + 6·47-s − 6·49-s − 6·51-s − 12·53-s + 5·57-s − 6·59-s − 13·61-s − 63-s + 11·67-s − 6·69-s + 2·73-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.38·13-s − 1.45·17-s + 1.14·19-s − 0.218·21-s − 1.25·23-s + 0.192·27-s − 1.11·29-s − 0.179·31-s + 1.04·33-s + 0.328·37-s + 0.800·39-s − 0.152·43-s + 0.875·47-s − 6/7·49-s − 0.840·51-s − 1.64·53-s + 0.662·57-s − 0.781·59-s − 1.66·61-s − 0.125·63-s + 1.34·67-s − 0.722·69-s + 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.627916149\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.627916149\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−11.66009069683484336461269580959, −10.93906516387019902338031029709, −9.496634055474639300529104381989, −9.108883986383451182036456380552, −8.026266654629362330508166642807, −6.77234120616327244201676192765, −6.03058890823729073510214550931, −4.24382547629296535183762439094, −3.43647697037916365579697730592, −1.63891885984744417728938078317,
1.63891885984744417728938078317, 3.43647697037916365579697730592, 4.24382547629296535183762439094, 6.03058890823729073510214550931, 6.77234120616327244201676192765, 8.026266654629362330508166642807, 9.108883986383451182036456380552, 9.496634055474639300529104381989, 10.93906516387019902338031029709, 11.66009069683484336461269580959