L(s) = 1 | + 2·5-s + 4·13-s + 8·17-s − 25-s − 10·29-s + 12·37-s − 8·41-s − 7·49-s + 14·53-s + 12·61-s + 8·65-s + 6·73-s + 16·85-s − 16·89-s + 18·97-s − 2·101-s + 20·109-s − 16·113-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 1.10·13-s + 1.94·17-s − 1/5·25-s − 1.85·29-s + 1.97·37-s − 1.24·41-s − 49-s + 1.92·53-s + 1.53·61-s + 0.992·65-s + 0.702·73-s + 1.73·85-s − 1.69·89-s + 1.82·97-s − 0.199·101-s + 1.91·109-s − 1.50·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.369296459\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.369296459\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−9.112200135449134934371783246207, −8.202694220619429288756420575978, −7.55985426853550376471644781916, −6.54863571929054381318876237423, −5.71994566936884971110657383524, −5.39262958273976329743454417116, −4.01100966495944443006006263063, −3.27092596459393225052086143179, −2.05899636987229362790422468605, −1.07574057656802494464390490749,
1.07574057656802494464390490749, 2.05899636987229362790422468605, 3.27092596459393225052086143179, 4.01100966495944443006006263063, 5.39262958273976329743454417116, 5.71994566936884971110657383524, 6.54863571929054381318876237423, 7.55985426853550376471644781916, 8.202694220619429288756420575978, 9.112200135449134934371783246207