| L(s) = 1 | + 4·3-s − 6·5-s − 11·9-s + 12·11-s + 82·13-s − 24·15-s + 30·17-s + 68·19-s − 216·23-s − 89·25-s − 152·27-s + 246·29-s − 112·31-s + 48·33-s + 110·37-s + 328·39-s + 246·41-s + 172·43-s + 66·45-s + 192·47-s + 120·51-s + 558·53-s − 72·55-s + 272·57-s + 540·59-s − 110·61-s − 492·65-s + ⋯ |
| L(s) = 1 | + 0.769·3-s − 0.536·5-s − 0.407·9-s + 0.328·11-s + 1.74·13-s − 0.413·15-s + 0.428·17-s + 0.821·19-s − 1.95·23-s − 0.711·25-s − 1.08·27-s + 1.57·29-s − 0.648·31-s + 0.253·33-s + 0.488·37-s + 1.34·39-s + 0.937·41-s + 0.609·43-s + 0.218·45-s + 0.595·47-s + 0.329·51-s + 1.44·53-s − 0.176·55-s + 0.632·57-s + 1.19·59-s − 0.230·61-s − 0.938·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.520565544\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.520565544\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 5 | \( 1 + 6 T + p^{3} T^{2} \) |
| 11 | \( 1 - 12 T + p^{3} T^{2} \) |
| 13 | \( 1 - 82 T + p^{3} T^{2} \) |
| 17 | \( 1 - 30 T + p^{3} T^{2} \) |
| 19 | \( 1 - 68 T + p^{3} T^{2} \) |
| 23 | \( 1 + 216 T + p^{3} T^{2} \) |
| 29 | \( 1 - 246 T + p^{3} T^{2} \) |
| 31 | \( 1 + 112 T + p^{3} T^{2} \) |
| 37 | \( 1 - 110 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 192 T + p^{3} T^{2} \) |
| 53 | \( 1 - 558 T + p^{3} T^{2} \) |
| 59 | \( 1 - 540 T + p^{3} T^{2} \) |
| 61 | \( 1 + 110 T + p^{3} T^{2} \) |
| 67 | \( 1 + 140 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 - 550 T + p^{3} T^{2} \) |
| 79 | \( 1 - 208 T + p^{3} T^{2} \) |
| 83 | \( 1 - 516 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1398 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1586 T + p^{3} T^{2} \) |
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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
−9.769034352446378686782239714300, −8.903191567931423043888874654542, −8.184564022024379779340210319988, −7.65921554575016039425677651638, −6.32377859642779558478157331807, −5.61251072491505662703388843050, −4.01316844088732895775534332057, −3.56354715961861568962582775665, −2.30122643760448740681340097053, −0.881032388084310648416644888712,
0.881032388084310648416644888712, 2.30122643760448740681340097053, 3.56354715961861568962582775665, 4.01316844088732895775534332057, 5.61251072491505662703388843050, 6.32377859642779558478157331807, 7.65921554575016039425677651638, 8.184564022024379779340210319988, 8.903191567931423043888874654542, 9.769034352446378686782239714300
