| L(s) = 1 | + 12·7-s − 64·11-s − 58·13-s + 32·17-s − 136·19-s − 128·23-s + 144·29-s + 20·31-s + 18·37-s + 288·41-s + 200·43-s + 384·47-s − 199·49-s + 496·53-s + 128·59-s − 458·61-s + 496·67-s − 512·71-s + 602·73-s − 768·77-s + 1.10e3·79-s + 704·83-s + 960·89-s − 696·91-s − 206·97-s − 432·101-s + 68·103-s + ⋯ |
| L(s) = 1 | + 0.647·7-s − 1.75·11-s − 1.23·13-s + 0.456·17-s − 1.64·19-s − 1.16·23-s + 0.922·29-s + 0.115·31-s + 0.0799·37-s + 1.09·41-s + 0.709·43-s + 1.19·47-s − 0.580·49-s + 1.28·53-s + 0.282·59-s − 0.961·61-s + 0.904·67-s − 0.855·71-s + 0.965·73-s − 1.13·77-s + 1.57·79-s + 0.931·83-s + 1.14·89-s − 0.801·91-s − 0.215·97-s − 0.425·101-s + 0.0650·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.354344641\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.354344641\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 12 T + p^{3} T^{2} \) |
| 11 | \( 1 + 64 T + p^{3} T^{2} \) |
| 13 | \( 1 + 58 T + p^{3} T^{2} \) |
| 17 | \( 1 - 32 T + p^{3} T^{2} \) |
| 19 | \( 1 + 136 T + p^{3} T^{2} \) |
| 23 | \( 1 + 128 T + p^{3} T^{2} \) |
| 29 | \( 1 - 144 T + p^{3} T^{2} \) |
| 31 | \( 1 - 20 T + p^{3} T^{2} \) |
| 37 | \( 1 - 18 T + p^{3} T^{2} \) |
| 41 | \( 1 - 288 T + p^{3} T^{2} \) |
| 43 | \( 1 - 200 T + p^{3} T^{2} \) |
| 47 | \( 1 - 384 T + p^{3} T^{2} \) |
| 53 | \( 1 - 496 T + p^{3} T^{2} \) |
| 59 | \( 1 - 128 T + p^{3} T^{2} \) |
| 61 | \( 1 + 458 T + p^{3} T^{2} \) |
| 67 | \( 1 - 496 T + p^{3} T^{2} \) |
| 71 | \( 1 + 512 T + p^{3} T^{2} \) |
| 73 | \( 1 - 602 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1108 T + p^{3} T^{2} \) |
| 83 | \( 1 - 704 T + p^{3} T^{2} \) |
| 89 | \( 1 - 960 T + p^{3} T^{2} \) |
| 97 | \( 1 + 206 T + p^{3} T^{2} \) |
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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
−8.800874112187846035429788234973, −7.87719622330779187695454105771, −7.70186843776768068231961500726, −6.50734158214462771843504520658, −5.56217980107796236009206546860, −4.86082614346935371369114056558, −4.09001050137307999689609026191, −2.62276207384847107767658971592, −2.15345934322336787199899964187, −0.51805784859413704271289582366,
0.51805784859413704271289582366, 2.15345934322336787199899964187, 2.62276207384847107767658971592, 4.09001050137307999689609026191, 4.86082614346935371369114056558, 5.56217980107796236009206546860, 6.50734158214462771843504520658, 7.70186843776768068231961500726, 7.87719622330779187695454105771, 8.800874112187846035429788234973
