L(s) = 1 | − 3-s − 2·5-s − 2·7-s + 9-s − 11-s − 6·13-s + 2·15-s − 4·17-s − 2·19-s + 2·21-s + 8·23-s − 25-s − 27-s + 33-s + 4·35-s + 6·37-s + 6·39-s + 10·43-s − 2·45-s − 3·49-s + 4·51-s − 14·53-s + 2·55-s + 2·57-s − 12·59-s + 14·61-s − 2·63-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.66·13-s + 0.516·15-s − 0.970·17-s − 0.458·19-s + 0.436·21-s + 1.66·23-s − 1/5·25-s − 0.192·27-s + 0.174·33-s + 0.676·35-s + 0.986·37-s + 0.960·39-s + 1.52·43-s − 0.298·45-s − 3/7·49-s + 0.560·51-s − 1.92·53-s + 0.269·55-s + 0.264·57-s − 1.56·59-s + 1.79·61-s − 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5630432314\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5630432314\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−9.307753379837837163372557025601, −8.198176922643024038087833917211, −7.39806002905685857271950948161, −6.86773830391836265877566482567, −6.01822078438710930204765558641, −4.90121038410657214500376601406, −4.42330001534322530762584667774, −3.26323990550872964687058288397, −2.32218584185209959702683831381, −0.47969970876592404988798780126,
0.47969970876592404988798780126, 2.32218584185209959702683831381, 3.26323990550872964687058288397, 4.42330001534322530762584667774, 4.90121038410657214500376601406, 6.01822078438710930204765558641, 6.86773830391836265877566482567, 7.39806002905685857271950948161, 8.198176922643024038087833917211, 9.307753379837837163372557025601