L(s) = 1 | − 3-s + 5-s + 1.64·7-s + 9-s − 0.354·11-s + 3.64·13-s − 15-s − 19-s − 1.64·21-s + 2·23-s + 25-s − 27-s + 9.64·29-s − 2·31-s + 0.354·33-s + 1.64·35-s − 6.93·37-s − 3.64·39-s + 1.64·41-s − 4.93·43-s + 45-s + 6·47-s − 4.29·49-s + 4·53-s − 0.354·55-s + 57-s + 3.29·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.622·7-s + 0.333·9-s − 0.106·11-s + 1.01·13-s − 0.258·15-s − 0.229·19-s − 0.359·21-s + 0.417·23-s + 0.200·25-s − 0.192·27-s + 1.79·29-s − 0.359·31-s + 0.0616·33-s + 0.278·35-s − 1.14·37-s − 0.583·39-s + 0.257·41-s − 0.752·43-s + 0.149·45-s + 0.875·47-s − 0.613·49-s + 0.549·53-s − 0.0477·55-s + 0.132·57-s + 0.428·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.834961892\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.834961892\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 - 1.64T + 7T^{2} \) |
| 11 | \( 1 + 0.354T + 11T^{2} \) |
| 13 | \( 1 - 3.64T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 - 2T + 23T^{2} \) |
| 29 | \( 1 - 9.64T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6.93T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 + 4.93T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 3.29T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 3.29T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 12.2T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−8.764025367129588370886465258356, −8.490206456789687055360650990514, −7.35099441995699186106102844355, −6.61236414002953121970011872076, −5.85830572743607915216842305942, −5.10988132792185754756427159550, −4.33593494186332130694837037782, −3.24372937489193266553420144693, −1.99202191841390697399070822072, −0.955359766689420643770804301787,
0.955359766689420643770804301787, 1.99202191841390697399070822072, 3.24372937489193266553420144693, 4.33593494186332130694837037782, 5.10988132792185754756427159550, 5.85830572743607915216842305942, 6.61236414002953121970011872076, 7.35099441995699186106102844355, 8.490206456789687055360650990514, 8.764025367129588370886465258356