L(s) = 1 | + 3-s − 2·4-s + 7-s + 9-s − 2·12-s + 13-s + 4·16-s + 6·17-s + 7·19-s + 21-s + 6·23-s + 27-s − 2·28-s + 6·29-s − 7·31-s − 2·36-s + 2·37-s + 39-s + 6·41-s + 43-s + 4·48-s − 6·49-s + 6·51-s − 2·52-s − 6·53-s + 7·57-s − 5·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 0.277·13-s + 16-s + 1.45·17-s + 1.60·19-s + 0.218·21-s + 1.25·23-s + 0.192·27-s − 0.377·28-s + 1.11·29-s − 1.25·31-s − 1/3·36-s + 0.328·37-s + 0.160·39-s + 0.937·41-s + 0.152·43-s + 0.577·48-s − 6/7·49-s + 0.840·51-s − 0.277·52-s − 0.824·53-s + 0.927·57-s − 0.640·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9075 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.627268870\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.627268870\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 7 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−7.63031263486409693645866170599, −7.49800821045644810342230388340, −6.29750798479964769166435801768, −5.43355079724659005816595726557, −5.01007605046464395690456689508, −4.21643506218253580349297559846, −3.31946827078264118724117246931, −2.98130430442646331703165247435, −1.49342337801977122111163171508, −0.863902230623839610208176326181,
0.863902230623839610208176326181, 1.49342337801977122111163171508, 2.98130430442646331703165247435, 3.31946827078264118724117246931, 4.21643506218253580349297559846, 5.01007605046464395690456689508, 5.43355079724659005816595726557, 6.29750798479964769166435801768, 7.49800821045644810342230388340, 7.63031263486409693645866170599