| L(s) = 1 | − 10·7-s + 46·11-s + 34·13-s + 66·17-s + 104·19-s + 164·23-s − 224·29-s − 72·31-s + 22·37-s − 194·41-s − 108·43-s − 480·47-s − 243·49-s + 286·53-s − 426·59-s + 698·61-s − 328·67-s − 188·71-s + 740·73-s − 460·77-s + 1.16e3·79-s + 412·83-s − 1.20e3·89-s − 340·91-s + 1.38e3·97-s + 1.12e3·101-s + 758·103-s + ⋯ |
| L(s) = 1 | − 0.539·7-s + 1.26·11-s + 0.725·13-s + 0.941·17-s + 1.25·19-s + 1.48·23-s − 1.43·29-s − 0.417·31-s + 0.0977·37-s − 0.738·41-s − 0.383·43-s − 1.48·47-s − 0.708·49-s + 0.741·53-s − 0.940·59-s + 1.46·61-s − 0.598·67-s − 0.314·71-s + 1.18·73-s − 0.680·77-s + 1.66·79-s + 0.544·83-s − 1.43·89-s − 0.391·91-s + 1.44·97-s + 1.11·101-s + 0.725·103-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
| \(L(2)\) |
\(\approx\) |
\(2.545927381\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.545927381\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 10 T + p^{3} T^{2} \) |
| 11 | \( 1 - 46 T + p^{3} T^{2} \) |
| 13 | \( 1 - 34 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 104 T + p^{3} T^{2} \) |
| 23 | \( 1 - 164 T + p^{3} T^{2} \) |
| 29 | \( 1 + 224 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 - 22 T + p^{3} T^{2} \) |
| 41 | \( 1 + 194 T + p^{3} T^{2} \) |
| 43 | \( 1 + 108 T + p^{3} T^{2} \) |
| 47 | \( 1 + 480 T + p^{3} T^{2} \) |
| 53 | \( 1 - 286 T + p^{3} T^{2} \) |
| 59 | \( 1 + 426 T + p^{3} T^{2} \) |
| 61 | \( 1 - 698 T + p^{3} T^{2} \) |
| 67 | \( 1 + 328 T + p^{3} T^{2} \) |
| 71 | \( 1 + 188 T + p^{3} T^{2} \) |
| 73 | \( 1 - 740 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1168 T + p^{3} T^{2} \) |
| 83 | \( 1 - 412 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1206 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1384 T + p^{3} T^{2} \) |
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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
−9.156513427124372929024229500569, −8.139498018739569821363195579803, −7.22981407113907119414112720152, −6.58700137605782178131103614809, −5.72158702073774228507234658495, −4.91105529698280620699530030695, −3.56082821096216276289696388279, −3.32076078718756507440224012841, −1.68335303615090477075765428687, −0.801561763487830935185335509736,
0.801561763487830935185335509736, 1.68335303615090477075765428687, 3.32076078718756507440224012841, 3.56082821096216276289696388279, 4.91105529698280620699530030695, 5.72158702073774228507234658495, 6.58700137605782178131103614809, 7.22981407113907119414112720152, 8.139498018739569821363195579803, 9.156513427124372929024229500569
