| L(s) = 1 | + 1.39·3-s − 4.73·7-s − 1.05·9-s − 2.34·11-s + 5.94·13-s + 3.32·17-s − 5.27·19-s − 6.61·21-s + 1.30·23-s − 5.65·27-s + 8.55·29-s + 2.25·31-s − 3.26·33-s − 4.99·37-s + 8.28·39-s + 4.43·41-s + 11.2·43-s − 2.67·47-s + 15.4·49-s + 4.63·51-s − 7.04·53-s − 7.35·57-s − 9.20·59-s + 4.90·61-s + 4.99·63-s + 0.532·67-s + 1.82·69-s + ⋯ |
| L(s) = 1 | + 0.805·3-s − 1.79·7-s − 0.351·9-s − 0.706·11-s + 1.64·13-s + 0.805·17-s − 1.20·19-s − 1.44·21-s + 0.272·23-s − 1.08·27-s + 1.58·29-s + 0.405·31-s − 0.569·33-s − 0.821·37-s + 1.32·39-s + 0.692·41-s + 1.71·43-s − 0.389·47-s + 2.20·49-s + 0.649·51-s − 0.968·53-s − 0.974·57-s − 1.19·59-s + 0.627·61-s + 0.629·63-s + 0.0651·67-s + 0.219·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.779310342\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.779310342\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 1.39T + 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 + 2.34T + 11T^{2} \) |
| 13 | \( 1 - 5.94T + 13T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 + 5.27T + 19T^{2} \) |
| 23 | \( 1 - 1.30T + 23T^{2} \) |
| 29 | \( 1 - 8.55T + 29T^{2} \) |
| 31 | \( 1 - 2.25T + 31T^{2} \) |
| 37 | \( 1 + 4.99T + 37T^{2} \) |
| 41 | \( 1 - 4.43T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + 2.67T + 47T^{2} \) |
| 53 | \( 1 + 7.04T + 53T^{2} \) |
| 59 | \( 1 + 9.20T + 59T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 - 0.532T + 67T^{2} \) |
| 71 | \( 1 - 11.7T + 71T^{2} \) |
| 73 | \( 1 + 9.65T + 73T^{2} \) |
| 79 | \( 1 - 17.0T + 79T^{2} \) |
| 83 | \( 1 + 6.13T + 83T^{2} \) |
| 89 | \( 1 - 1.26T + 89T^{2} \) |
| 97 | \( 1 + 0.171T + 97T^{2} \) |
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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
−8.487839926213153652588316069247, −7.935341263479288811901098497282, −6.91844169539335273931749417198, −6.15348875738991098450507279257, −5.82243425065933426955955390573, −4.48204395387296497291666863838, −3.45803555726512465564592587075, −3.15243835068911931001873156799, −2.26414395251734939994870053320, −0.71025139065869643104829654737,
0.71025139065869643104829654737, 2.26414395251734939994870053320, 3.15243835068911931001873156799, 3.45803555726512465564592587075, 4.48204395387296497291666863838, 5.82243425065933426955955390573, 6.15348875738991098450507279257, 6.91844169539335273931749417198, 7.935341263479288811901098497282, 8.487839926213153652588316069247
