| L(s) = 1 | + 2·3-s − 3·5-s + 7-s + 9-s + 3·11-s − 4·13-s − 6·15-s − 3·17-s + 2·21-s + 4·25-s − 4·27-s + 6·29-s − 4·31-s + 6·33-s − 3·35-s + 2·37-s − 8·39-s + 6·41-s − 43-s − 3·45-s + 3·47-s − 6·49-s − 6·51-s + 12·53-s − 9·55-s + 6·59-s + 61-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1.34·5-s + 0.377·7-s + 1/3·9-s + 0.904·11-s − 1.10·13-s − 1.54·15-s − 0.727·17-s + 0.436·21-s + 4/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s + 1.04·33-s − 0.507·35-s + 0.328·37-s − 1.28·39-s + 0.937·41-s − 0.152·43-s − 0.447·45-s + 0.437·47-s − 6/7·49-s − 0.840·51-s + 1.64·53-s − 1.21·55-s + 0.781·59-s + 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−15.49082619855995, −15.10220662961855, −14.72961832043553, −14.22994804246405, −13.81671635649156, −13.02047431934341, −12.47887655746177, −11.87496347045407, −11.50433493716809, −10.96752708752513, −10.16617354122306, −9.500173836378064, −8.976678032869397, −8.483758953567513, −7.993484740425229, −7.440271850244603, −7.002990865534729, −6.264941883276721, −5.253514283881998, −4.579866266677584, −3.965070056488299, −3.594342304429923, −2.649984530936166, −2.221456140337860, −1.081811256145575, 0,
1.081811256145575, 2.221456140337860, 2.649984530936166, 3.594342304429923, 3.965070056488299, 4.579866266677584, 5.253514283881998, 6.264941883276721, 7.002990865534729, 7.440271850244603, 7.993484740425229, 8.483758953567513, 8.976678032869397, 9.500173836378064, 10.16617354122306, 10.96752708752513, 11.50433493716809, 11.87496347045407, 12.47887655746177, 13.02047431934341, 13.81671635649156, 14.22994804246405, 14.72961832043553, 15.10220662961855, 15.49082619855995
