| L(s) = 1 | − 5·5-s − 7·7-s − 68·11-s + 22·13-s + 30·17-s + 108·19-s − 184·23-s + 25·25-s − 166·29-s − 32·31-s + 35·35-s − 370·37-s − 154·41-s + 212·43-s + 512·47-s + 49·49-s + 98·53-s + 340·55-s + 860·59-s + 390·61-s − 110·65-s + 60·67-s − 840·71-s − 630·73-s + 476·77-s + 1.31e3·79-s + 436·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 1.86·11-s + 0.469·13-s + 0.428·17-s + 1.30·19-s − 1.66·23-s + 1/5·25-s − 1.06·29-s − 0.185·31-s + 0.169·35-s − 1.64·37-s − 0.586·41-s + 0.751·43-s + 1.58·47-s + 1/7·49-s + 0.253·53-s + 0.833·55-s + 1.89·59-s + 0.818·61-s − 0.209·65-s + 0.109·67-s − 1.40·71-s − 1.01·73-s + 0.704·77-s + 1.86·79-s + 0.576·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1260 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.145919866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.145919866\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + p T \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 + 68 T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 108 T + p^{3} T^{2} \) |
| 23 | \( 1 + 8 p T + p^{3} T^{2} \) |
| 29 | \( 1 + 166 T + p^{3} T^{2} \) |
| 31 | \( 1 + 32 T + p^{3} T^{2} \) |
| 37 | \( 1 + 10 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 154 T + p^{3} T^{2} \) |
| 43 | \( 1 - 212 T + p^{3} T^{2} \) |
| 47 | \( 1 - 512 T + p^{3} T^{2} \) |
| 53 | \( 1 - 98 T + p^{3} T^{2} \) |
| 59 | \( 1 - 860 T + p^{3} T^{2} \) |
| 61 | \( 1 - 390 T + p^{3} T^{2} \) |
| 67 | \( 1 - 60 T + p^{3} T^{2} \) |
| 71 | \( 1 + 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 630 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1312 T + p^{3} T^{2} \) |
| 83 | \( 1 - 436 T + p^{3} T^{2} \) |
| 89 | \( 1 - 598 T + p^{3} T^{2} \) |
| 97 | \( 1 - 914 T + p^{3} T^{2} \) |
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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
−9.359135200039088205751170145814, −8.345779022944018750933216413319, −7.70679283816717902774315902682, −7.06125327039218073266996664290, −5.70437827577319710603462253761, −5.32721997043755681976283447233, −3.98358766977492811244893421930, −3.18526267764480253228628010100, −2.09299825982544980392784881641, −0.52113604024001734681359538122,
0.52113604024001734681359538122, 2.09299825982544980392784881641, 3.18526267764480253228628010100, 3.98358766977492811244893421930, 5.32721997043755681976283447233, 5.70437827577319710603462253761, 7.06125327039218073266996664290, 7.70679283816717902774315902682, 8.345779022944018750933216413319, 9.359135200039088205751170145814
