| L(s) = 1 | + 3-s − 2·5-s + 4·7-s + 9-s − 11-s + 6·13-s − 2·15-s + 6·17-s − 8·19-s + 4·21-s − 25-s + 27-s − 6·29-s − 33-s − 8·35-s + 6·37-s + 6·39-s − 10·41-s − 8·43-s − 2·45-s + 9·49-s + 6·51-s + 6·53-s + 2·55-s − 8·57-s + 4·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 0.301·11-s + 1.66·13-s − 0.516·15-s + 1.45·17-s − 1.83·19-s + 0.872·21-s − 1/5·25-s + 0.192·27-s − 1.11·29-s − 0.174·33-s − 1.35·35-s + 0.986·37-s + 0.960·39-s − 1.56·41-s − 1.21·43-s − 0.298·45-s + 9/7·49-s + 0.840·51-s + 0.824·53-s + 0.269·55-s − 1.05·57-s + 0.520·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.538171885\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.538171885\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−11.76915955329869490716285473842, −11.13257530604792818900470942011, −10.22105383755109362423503001213, −8.570379528880000367723152717773, −8.281506035956720263967432968159, −7.38236864490209309055406166401, −5.83918238198525237740190555867, −4.46595770042798361252405180269, −3.53679289542094103742671365346, −1.66684278577423700169699871962,
1.66684278577423700169699871962, 3.53679289542094103742671365346, 4.46595770042798361252405180269, 5.83918238198525237740190555867, 7.38236864490209309055406166401, 8.281506035956720263967432968159, 8.570379528880000367723152717773, 10.22105383755109362423503001213, 11.13257530604792818900470942011, 11.76915955329869490716285473842
