| L(s) = 1 | + 20·5-s + 32·7-s + 50·11-s − 13·13-s + 30·17-s + 120·19-s − 20·23-s + 275·25-s − 82·29-s + 44·31-s + 640·35-s − 306·37-s − 108·41-s + 356·43-s − 178·47-s + 681·49-s − 198·53-s + 1.00e3·55-s + 94·59-s − 62·61-s − 260·65-s + 140·67-s − 778·71-s + 62·73-s + 1.60e3·77-s + 1.09e3·79-s − 462·83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 1.72·7-s + 1.37·11-s − 0.277·13-s + 0.428·17-s + 1.44·19-s − 0.181·23-s + 11/5·25-s − 0.525·29-s + 0.254·31-s + 3.09·35-s − 1.35·37-s − 0.411·41-s + 1.26·43-s − 0.552·47-s + 1.98·49-s − 0.513·53-s + 2.45·55-s + 0.207·59-s − 0.130·61-s − 0.496·65-s + 0.255·67-s − 1.30·71-s + 0.0994·73-s + 2.36·77-s + 1.56·79-s − 0.610·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1872 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.976586363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.976586363\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 + p T \) |
| good | 5 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 7 | \( 1 - 32 T + p^{3} T^{2} \) |
| 11 | \( 1 - 50 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 120 T + p^{3} T^{2} \) |
| 23 | \( 1 + 20 T + p^{3} T^{2} \) |
| 29 | \( 1 + 82 T + p^{3} T^{2} \) |
| 31 | \( 1 - 44 T + p^{3} T^{2} \) |
| 37 | \( 1 + 306 T + p^{3} T^{2} \) |
| 41 | \( 1 + 108 T + p^{3} T^{2} \) |
| 43 | \( 1 - 356 T + p^{3} T^{2} \) |
| 47 | \( 1 + 178 T + p^{3} T^{2} \) |
| 53 | \( 1 + 198 T + p^{3} T^{2} \) |
| 59 | \( 1 - 94 T + p^{3} T^{2} \) |
| 61 | \( 1 + 62 T + p^{3} T^{2} \) |
| 67 | \( 1 - 140 T + p^{3} T^{2} \) |
| 71 | \( 1 + 778 T + p^{3} T^{2} \) |
| 73 | \( 1 - 62 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1096 T + p^{3} T^{2} \) |
| 83 | \( 1 + 462 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1224 T + p^{3} T^{2} \) |
| 97 | \( 1 - 614 T + p^{3} T^{2} \) |
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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
−9.080465143724011107646840686881, −8.153336838660082380586471700305, −7.25537867070882894371418089846, −6.42538971584480242717742637659, −5.44031705224444940141550033455, −5.14121441353233059492200685150, −3.99560880300017240192552094845, −2.66870356847878150407984676909, −1.54507586180003651196671517905, −1.31646195162867333974558022299,
1.31646195162867333974558022299, 1.54507586180003651196671517905, 2.66870356847878150407984676909, 3.99560880300017240192552094845, 5.14121441353233059492200685150, 5.44031705224444940141550033455, 6.42538971584480242717742637659, 7.25537867070882894371418089846, 8.153336838660082380586471700305, 9.080465143724011107646840686881
