| L(s) = 1 | + 2·3-s − 7-s + 9-s + 4·11-s + 6·17-s − 6·19-s − 2·21-s + 23-s − 5·25-s − 4·27-s − 10·29-s − 4·31-s + 8·33-s + 2·37-s − 10·41-s − 4·43-s − 12·47-s + 49-s + 12·51-s + 6·53-s − 12·57-s − 2·59-s − 63-s + 2·69-s + 8·71-s − 6·73-s − 10·75-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.45·17-s − 1.37·19-s − 0.436·21-s + 0.208·23-s − 25-s − 0.769·27-s − 1.85·29-s − 0.718·31-s + 1.39·33-s + 0.328·37-s − 1.56·41-s − 0.609·43-s − 1.75·47-s + 1/7·49-s + 1.68·51-s + 0.824·53-s − 1.58·57-s − 0.260·59-s − 0.125·63-s + 0.240·69-s + 0.949·71-s − 0.702·73-s − 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 2 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−16.82932825791711, −16.53627012674020, −15.48630555173004, −14.93841290373823, −14.69942615469793, −14.12081664854446, −13.50528016643797, −12.94575810266352, −12.41935582126153, −11.55782353717944, −11.26379153059736, −10.12418726450732, −9.794271656572296, −9.148093868475362, −8.632941414197224, −8.037277722996302, −7.405673891886393, −6.708379789577481, −5.980249573980565, −5.330108524546335, −4.200030024740632, −3.602095908503061, −3.227972568938517, −2.079132253846368, −1.551615123121714, 0,
1.551615123121714, 2.079132253846368, 3.227972568938517, 3.602095908503061, 4.200030024740632, 5.330108524546335, 5.980249573980565, 6.708379789577481, 7.405673891886393, 8.037277722996302, 8.632941414197224, 9.148093868475362, 9.794271656572296, 10.12418726450732, 11.26379153059736, 11.55782353717944, 12.41935582126153, 12.94575810266352, 13.50528016643797, 14.12081664854446, 14.69942615469793, 14.93841290373823, 15.48630555173004, 16.53627012674020, 16.82932825791711
