| L(s) = 1 | + 3-s + 5·5-s + 32·7-s − 26·9-s + 30·11-s + 19·13-s + 5·15-s − 60·17-s + 58·19-s + 32·21-s − 23·23-s + 25·25-s − 53·27-s + 85·29-s + 65·31-s + 30·33-s + 160·35-s − 34·37-s + 19·39-s + 143·41-s + 332·43-s − 130·45-s + 561·47-s + 681·49-s − 60·51-s − 422·53-s + 150·55-s + ⋯ |
| L(s) = 1 | + 0.192·3-s + 0.447·5-s + 1.72·7-s − 0.962·9-s + 0.822·11-s + 0.405·13-s + 0.0860·15-s − 0.856·17-s + 0.700·19-s + 0.332·21-s − 0.208·23-s + 1/5·25-s − 0.377·27-s + 0.544·29-s + 0.376·31-s + 0.158·33-s + 0.772·35-s − 0.151·37-s + 0.0780·39-s + 0.544·41-s + 1.17·43-s − 0.430·45-s + 1.74·47-s + 1.98·49-s − 0.164·51-s − 1.09·53-s + 0.367·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.453462533\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.453462533\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - p T \) |
| 23 | \( 1 + p T \) |
| good | 3 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 30 T + p^{3} T^{2} \) |
| 13 | \( 1 - 19 T + p^{3} T^{2} \) |
| 17 | \( 1 + 60 T + p^{3} T^{2} \) |
| 19 | \( 1 - 58 T + p^{3} T^{2} \) |
| 29 | \( 1 - 85 T + p^{3} T^{2} \) |
| 31 | \( 1 - 65 T + p^{3} T^{2} \) |
| 37 | \( 1 + 34 T + p^{3} T^{2} \) |
| 41 | \( 1 - 143 T + p^{3} T^{2} \) |
| 43 | \( 1 - 332 T + p^{3} T^{2} \) |
| 47 | \( 1 - 561 T + p^{3} T^{2} \) |
| 53 | \( 1 + 422 T + p^{3} T^{2} \) |
| 59 | \( 1 + 392 T + p^{3} T^{2} \) |
| 61 | \( 1 + 246 T + p^{3} T^{2} \) |
| 67 | \( 1 + 894 T + p^{3} T^{2} \) |
| 71 | \( 1 - 737 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1041 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1114 T + p^{3} T^{2} \) |
| 83 | \( 1 - 936 T + p^{3} T^{2} \) |
| 89 | \( 1 - 824 T + p^{3} T^{2} \) |
| 97 | \( 1 + 868 T + p^{3} T^{2} \) |
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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
−8.907659013871024295625418555609, −8.168981806390034961771511967804, −7.50132476529534977659829415326, −6.38892627740135222072948232582, −5.66500515685318688609639389052, −4.81077116996280810788760225782, −4.02952132633300369118540734099, −2.76094642827822378265928538778, −1.84862534409596015138624299216, −0.912184399092809719541867253753,
0.912184399092809719541867253753, 1.84862534409596015138624299216, 2.76094642827822378265928538778, 4.02952132633300369118540734099, 4.81077116996280810788760225782, 5.66500515685318688609639389052, 6.38892627740135222072948232582, 7.50132476529534977659829415326, 8.168981806390034961771511967804, 8.907659013871024295625418555609
