| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s − 12·7-s + 8·8-s − 10·10-s + 20·11-s − 4·13-s − 24·14-s + 16·16-s + 34·17-s − 19·19-s − 20·20-s + 40·22-s − 40·23-s + 25·25-s − 8·26-s − 48·28-s + 150·29-s − 200·31-s + 32·32-s + 68·34-s + 60·35-s − 156·37-s − 38·38-s − 40·40-s + 218·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.647·7-s + 0.353·8-s − 0.316·10-s + 0.548·11-s − 0.0853·13-s − 0.458·14-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.223·20-s + 0.387·22-s − 0.362·23-s + 1/5·25-s − 0.0603·26-s − 0.323·28-s + 0.960·29-s − 1.15·31-s + 0.176·32-s + 0.342·34-s + 0.289·35-s − 0.693·37-s − 0.162·38-s − 0.158·40-s + 0.830·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.883120821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.883120821\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 19 | \( 1 + p T \) |
| good | 7 | \( 1 + 12 T + p^{3} T^{2} \) |
| 11 | \( 1 - 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 4 T + p^{3} T^{2} \) |
| 17 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 40 T + p^{3} T^{2} \) |
| 29 | \( 1 - 150 T + p^{3} T^{2} \) |
| 31 | \( 1 + 200 T + p^{3} T^{2} \) |
| 37 | \( 1 + 156 T + p^{3} T^{2} \) |
| 41 | \( 1 - 218 T + p^{3} T^{2} \) |
| 43 | \( 1 - 248 T + p^{3} T^{2} \) |
| 47 | \( 1 - 180 T + p^{3} T^{2} \) |
| 53 | \( 1 + 72 T + p^{3} T^{2} \) |
| 59 | \( 1 - 48 T + p^{3} T^{2} \) |
| 61 | \( 1 + 134 T + p^{3} T^{2} \) |
| 67 | \( 1 - 334 T + p^{3} T^{2} \) |
| 71 | \( 1 - 520 T + p^{3} T^{2} \) |
| 73 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 79 | \( 1 - 980 T + p^{3} T^{2} \) |
| 83 | \( 1 - 156 T + p^{3} T^{2} \) |
| 89 | \( 1 + 670 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1124 T + p^{3} T^{2} \) |
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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
−9.005931276242756128426421359981, −8.043410327340732785660417470321, −7.24071102820727950657996511104, −6.49502234337747324010027978649, −5.75620798054966070149143388619, −4.77219515350107409598550533706, −3.87139763362150186274536145076, −3.22325169199478297946587198865, −2.08498127444427473241654092914, −0.71808332875396461020556546743,
0.71808332875396461020556546743, 2.08498127444427473241654092914, 3.22325169199478297946587198865, 3.87139763362150186274536145076, 4.77219515350107409598550533706, 5.75620798054966070149143388619, 6.49502234337747324010027978649, 7.24071102820727950657996511104, 8.043410327340732785660417470321, 9.005931276242756128426421359981
