| L(s) = 1 | − 4·3-s − 5·5-s + 7·7-s − 11·9-s + 68·11-s − 22·13-s + 20·15-s − 30·17-s + 108·19-s − 28·21-s − 184·23-s + 25·25-s + 152·27-s − 166·29-s + 32·31-s − 272·33-s − 35·35-s + 370·37-s + 88·39-s + 154·41-s + 212·43-s + 55·45-s + 512·47-s + 49·49-s + 120·51-s + 98·53-s − 340·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 0.447·5-s + 0.377·7-s − 0.407·9-s + 1.86·11-s − 0.469·13-s + 0.344·15-s − 0.428·17-s + 1.30·19-s − 0.290·21-s − 1.66·23-s + 1/5·25-s + 1.08·27-s − 1.06·29-s + 0.185·31-s − 1.43·33-s − 0.169·35-s + 1.64·37-s + 0.361·39-s + 0.586·41-s + 0.751·43-s + 0.182·45-s + 1.58·47-s + 1/7·49-s + 0.329·51-s + 0.253·53-s − 0.833·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.403459479\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.403459479\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 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
−8.831656361494078023867014969701, −7.72448905109604314954914435055, −7.21518370955627074059441458377, −6.05213608534607902027014374076, −5.82995188728438513239828821009, −4.52381905168687485915744382438, −4.06025210662036937178881216831, −2.88320168442251166314512539658, −1.57798927369772450663654532349, −0.57952907432344938765859853364,
0.57952907432344938765859853364, 1.57798927369772450663654532349, 2.88320168442251166314512539658, 4.06025210662036937178881216831, 4.52381905168687485915744382438, 5.82995188728438513239828821009, 6.05213608534607902027014374076, 7.21518370955627074059441458377, 7.72448905109604314954914435055, 8.831656361494078023867014969701
