| L(s) = 1 | − 3-s + 7-s + 9-s + 2·13-s + 6·17-s − 4·19-s − 21-s − 27-s + 6·29-s + 4·31-s + 2·37-s − 2·39-s + 6·41-s − 8·43-s − 12·47-s + 49-s − 6·51-s + 6·53-s + 4·57-s − 12·59-s − 2·61-s + 63-s − 8·67-s − 14·73-s + 16·79-s + 81-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 1.21·43-s − 1.75·47-s + 1/7·49-s − 0.840·51-s + 0.824·53-s + 0.529·57-s − 1.56·59-s − 0.256·61-s + 0.125·63-s − 0.977·67-s − 1.63·73-s + 1.80·79-s + 1/9·81-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.19435013075614, −14.74079063539729, −14.32255792657175, −13.52760976331861, −13.27374096812870, −12.46632253832686, −12.04194016263953, −11.69736931167879, −10.90345543182341, −10.58909730566052, −10.00801397089079, −9.476452783624830, −8.739298554019674, −8.031433596981929, −7.898556427725120, −6.901058693766526, −6.458002341338438, −5.872004593082327, −5.306007953760068, −4.614096812217202, −4.146399435665722, −3.273175048766160, −2.677493388614919, −1.578154585296770, −1.114408695829257, 0,
1.114408695829257, 1.578154585296770, 2.677493388614919, 3.273175048766160, 4.146399435665722, 4.614096812217202, 5.306007953760068, 5.872004593082327, 6.458002341338438, 6.901058693766526, 7.898556427725120, 8.031433596981929, 8.739298554019674, 9.476452783624830, 10.00801397089079, 10.58909730566052, 10.90345543182341, 11.69736931167879, 12.04194016263953, 12.46632253832686, 13.27374096812870, 13.52760976331861, 14.32255792657175, 14.74079063539729, 15.19435013075614
