| L(s) = 1 | + 3·3-s − 5·5-s + 9·9-s − 4·11-s − 54·13-s − 15·15-s + 114·17-s − 44·19-s + 96·23-s + 25·25-s + 27·27-s − 134·29-s − 272·31-s − 12·33-s + 98·37-s − 162·39-s − 6·41-s − 12·43-s − 45·45-s − 200·47-s − 343·49-s + 342·51-s − 654·53-s + 20·55-s − 132·57-s − 36·59-s + 442·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.109·11-s − 1.15·13-s − 0.258·15-s + 1.62·17-s − 0.531·19-s + 0.870·23-s + 1/5·25-s + 0.192·27-s − 0.858·29-s − 1.57·31-s − 0.0633·33-s + 0.435·37-s − 0.665·39-s − 0.0228·41-s − 0.0425·43-s − 0.149·45-s − 0.620·47-s − 49-s + 0.939·51-s − 1.69·53-s + 0.0490·55-s − 0.306·57-s − 0.0794·59-s + 0.927·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 4 T + p^{3} T^{2} \) |
| 13 | \( 1 + 54 T + p^{3} T^{2} \) |
| 17 | \( 1 - 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 134 T + p^{3} T^{2} \) |
| 31 | \( 1 + 272 T + p^{3} T^{2} \) |
| 37 | \( 1 - 98 T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 12 T + p^{3} T^{2} \) |
| 47 | \( 1 + 200 T + p^{3} T^{2} \) |
| 53 | \( 1 + 654 T + p^{3} T^{2} \) |
| 59 | \( 1 + 36 T + p^{3} T^{2} \) |
| 61 | \( 1 - 442 T + p^{3} T^{2} \) |
| 67 | \( 1 - 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 632 T + p^{3} T^{2} \) |
| 73 | \( 1 + 390 T + p^{3} T^{2} \) |
| 79 | \( 1 - 688 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 + 694 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1726 T + p^{3} T^{2} \) |
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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
−9.334819565463355959467511007584, −8.269786707614634640318832920033, −7.60051526523893229330377928361, −6.95636593583712891378760299789, −5.63491239674832032803610544620, −4.77037217176334062310418693381, −3.65629098082578879074948510161, −2.82699907670576057327406181500, −1.54815362567872107374973424293, 0,
1.54815362567872107374973424293, 2.82699907670576057327406181500, 3.65629098082578879074948510161, 4.77037217176334062310418693381, 5.63491239674832032803610544620, 6.95636593583712891378760299789, 7.60051526523893229330377928361, 8.269786707614634640318832920033, 9.334819565463355959467511007584
