L(s) = 1 | − 3-s − 5-s − 7-s + 9-s + 2·13-s + 15-s + 6·17-s + 4·19-s + 21-s + 23-s + 25-s − 27-s + 6·29-s + 4·31-s + 35-s + 2·37-s − 2·39-s − 6·41-s − 8·43-s − 45-s − 12·47-s + 49-s − 6·51-s − 6·53-s − 4·57-s + 12·59-s + 2·61-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.377·7-s + 1/3·9-s + 0.554·13-s + 0.258·15-s + 1.45·17-s + 0.917·19-s + 0.218·21-s + 0.208·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.169·35-s + 0.328·37-s − 0.320·39-s − 0.937·41-s − 1.21·43-s − 0.149·45-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.04381238626102, −14.66646024358993, −14.01597875773403, −13.44356214535970, −13.03098100435615, −12.29973184041076, −11.95565437422674, −11.51392550567162, −11.00609111565719, −10.15503604261197, −9.990195150691630, −9.439407881493878, −8.398412597022782, −8.292003143560617, −7.505788457101379, −6.896204985055149, −6.434985125915477, −5.780997121895723, −5.169057856549361, −4.702476361391360, −3.841380108449324, −3.273685868901619, −2.794869571995765, −1.494217955118105, −1.019320848805071, 0,
1.019320848805071, 1.494217955118105, 2.794869571995765, 3.273685868901619, 3.841380108449324, 4.702476361391360, 5.169057856549361, 5.780997121895723, 6.434985125915477, 6.896204985055149, 7.505788457101379, 8.292003143560617, 8.398412597022782, 9.439407881493878, 9.990195150691630, 10.15503604261197, 11.00609111565719, 11.51392550567162, 11.95565437422674, 12.29973184041076, 13.03098100435615, 13.44356214535970, 14.01597875773403, 14.66646024358993, 15.04381238626102