| L(s) = 1 | − 2·2-s − 6.13·3-s + 4·4-s + 12.2·6-s + 18.5·7-s − 8·8-s + 10.6·9-s − 15.3·11-s − 24.5·12-s − 27.8·13-s − 37.0·14-s + 16·16-s + 62.4·17-s − 21.2·18-s + 55.3·19-s − 113.·21-s + 30.7·22-s − 44.3·23-s + 49.0·24-s + 55.6·26-s + 100.·27-s + 74.0·28-s + 29·29-s − 207.·31-s − 32·32-s + 94.3·33-s − 124.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.18·3-s + 0.5·4-s + 0.834·6-s + 0.999·7-s − 0.353·8-s + 0.393·9-s − 0.421·11-s − 0.590·12-s − 0.593·13-s − 0.706·14-s + 0.250·16-s + 0.890·17-s − 0.277·18-s + 0.668·19-s − 1.17·21-s + 0.298·22-s − 0.401·23-s + 0.417·24-s + 0.419·26-s + 0.716·27-s + 0.499·28-s + 0.185·29-s − 1.19·31-s − 0.176·32-s + 0.497·33-s − 0.629·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 29 | \( 1 - 29T \) |
| good | 3 | \( 1 + 6.13T + 27T^{2} \) |
| 7 | \( 1 - 18.5T + 343T^{2} \) |
| 11 | \( 1 + 15.3T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 62.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 55.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 44.3T + 1.21e4T^{2} \) |
| 31 | \( 1 + 207.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 125.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 101.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 50.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 692.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 557.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 809.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 749.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 54.7T + 3.57e5T^{2} \) |
| 73 | \( 1 - 184.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 752.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 902.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 953.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.11e3T + 9.12e5T^{2} \) |
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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
−8.689867753607866212556049489753, −7.88310729417381580896897597811, −7.26177830869537071754371278355, −6.29950594922726266123757763559, −5.29475702675312795828512938777, −5.02446808320722911693101031007, −3.51400058002200928613026314550, −2.14995617991649607972031687394, −1.07003346059515082832028977355, 0,
1.07003346059515082832028977355, 2.14995617991649607972031687394, 3.51400058002200928613026314550, 5.02446808320722911693101031007, 5.29475702675312795828512938777, 6.29950594922726266123757763559, 7.26177830869537071754371278355, 7.88310729417381580896897597811, 8.689867753607866212556049489753
