L(s) = 1 | + 1.73·5-s − 7-s + 1.73·11-s − 1.73·17-s + 19-s + 1.99·25-s − 1.73·35-s + 43-s − 1.73·47-s + 2.99·55-s + 61-s − 73-s − 1.73·77-s − 2.99·85-s + 1.73·95-s + 1.73·119-s + ⋯ |
L(s) = 1 | + 1.73·5-s − 7-s + 1.73·11-s − 1.73·17-s + 19-s + 1.99·25-s − 1.73·35-s + 43-s − 1.73·47-s + 2.99·55-s + 61-s − 73-s − 1.73·77-s − 2.99·85-s + 1.73·95-s + 1.73·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.620659815\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.620659815\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 - 1.73T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + 1.73T + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T + T^{2} \) |
| 47 | \( 1 + 1.73T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.235990735237979540840943919229, −8.619410630756537695021763134250, −7.15794941416174872333860017999, −6.46591072558838029086479624716, −6.23514015606536678206407764949, −5.26621118161322420950977046831, −4.27314550724798512049580917546, −3.25098285817593569644123243872, −2.27302423973733520947508430752, −1.33300774873847751313801934917,
1.33300774873847751313801934917, 2.27302423973733520947508430752, 3.25098285817593569644123243872, 4.27314550724798512049580917546, 5.26621118161322420950977046831, 6.23514015606536678206407764949, 6.46591072558838029086479624716, 7.15794941416174872333860017999, 8.619410630756537695021763134250, 9.235990735237979540840943919229