| L(s) = 1 | − 14·5-s − 7·7-s − 4·11-s + 54·13-s + 14·17-s + 92·19-s + 152·23-s + 71·25-s + 106·29-s − 144·31-s + 98·35-s + 158·37-s + 390·41-s − 508·43-s + 528·47-s + 49·49-s − 606·53-s + 56·55-s + 364·59-s + 678·61-s − 756·65-s + 844·67-s + 8·71-s − 422·73-s + 28·77-s + 384·79-s + 548·83-s + ⋯ |
| L(s) = 1 | − 1.25·5-s − 0.377·7-s − 0.109·11-s + 1.15·13-s + 0.199·17-s + 1.11·19-s + 1.37·23-s + 0.567·25-s + 0.678·29-s − 0.834·31-s + 0.473·35-s + 0.702·37-s + 1.48·41-s − 1.80·43-s + 1.63·47-s + 1/7·49-s − 1.57·53-s + 0.137·55-s + 0.803·59-s + 1.42·61-s − 1.44·65-s + 1.53·67-s + 0.0133·71-s − 0.676·73-s + 0.0414·77-s + 0.546·79-s + 0.724·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.372920706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.372920706\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + p T \) |
| good | 5 | \( 1 + 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 - 92 T + p^{3} T^{2} \) |
| 23 | \( 1 - 152 T + p^{3} T^{2} \) |
| 29 | \( 1 - 106 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 - 158 T + p^{3} T^{2} \) |
| 41 | \( 1 - 390 T + p^{3} T^{2} \) |
| 43 | \( 1 + 508 T + p^{3} T^{2} \) |
| 47 | \( 1 - 528 T + p^{3} T^{2} \) |
| 53 | \( 1 + 606 T + p^{3} T^{2} \) |
| 59 | \( 1 - 364 T + p^{3} T^{2} \) |
| 61 | \( 1 - 678 T + p^{3} T^{2} \) |
| 67 | \( 1 - 844 T + p^{3} T^{2} \) |
| 71 | \( 1 - 8 T + p^{3} T^{2} \) |
| 73 | \( 1 + 422 T + p^{3} T^{2} \) |
| 79 | \( 1 - 384 T + p^{3} T^{2} \) |
| 83 | \( 1 - 548 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1194 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1502 T + p^{3} T^{2} \) |
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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
−11.48016647561398649217381205683, −10.92664613291796964048540556198, −9.619435359985852412357928789070, −8.586336500963135596512388445969, −7.68576081818539569826698250945, −6.73689945832073349827511786316, −5.38967628174797762816757374268, −4.02492502434093766009312278552, −3.09962663529711795514319748122, −0.871160386801707880068417972706,
0.871160386801707880068417972706, 3.09962663529711795514319748122, 4.02492502434093766009312278552, 5.38967628174797762816757374268, 6.73689945832073349827511786316, 7.68576081818539569826698250945, 8.586336500963135596512388445969, 9.619435359985852412357928789070, 10.92664613291796964048540556198, 11.48016647561398649217381205683
