| L(s) = 1 | + 5.40e4·3-s + 1.95e6·5-s − 4.83e7·7-s + 1.76e9·9-s + 1.12e10·11-s − 1.22e10·13-s + 1.05e11·15-s − 6.70e11·17-s − 1.82e12·19-s − 2.61e12·21-s − 1.15e13·23-s + 3.81e12·25-s + 3.25e13·27-s − 8.95e13·29-s − 2.42e14·31-s + 6.07e14·33-s − 9.44e13·35-s + 8.35e14·37-s − 6.64e14·39-s + 4.03e14·41-s − 2.07e15·43-s + 3.44e15·45-s − 1.47e15·47-s − 9.06e15·49-s − 3.62e16·51-s − 2.12e16·53-s + 2.19e16·55-s + ⋯ |
| L(s) = 1 | + 1.58·3-s + 0.447·5-s − 0.452·7-s + 1.51·9-s + 1.43·11-s − 0.321·13-s + 0.709·15-s − 1.37·17-s − 1.29·19-s − 0.718·21-s − 1.33·23-s + 0.199·25-s + 0.820·27-s − 1.14·29-s − 1.64·31-s + 2.27·33-s − 0.202·35-s + 1.05·37-s − 0.510·39-s + 0.192·41-s − 0.629·43-s + 0.678·45-s − 0.192·47-s − 0.795·49-s − 2.17·51-s − 0.883·53-s + 0.642·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 1.95e6T \) |
| good | 3 | \( 1 - 5.40e4T + 1.16e9T^{2} \) |
| 7 | \( 1 + 4.83e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 1.12e10T + 6.11e19T^{2} \) |
| 13 | \( 1 + 1.22e10T + 1.46e21T^{2} \) |
| 17 | \( 1 + 6.70e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 1.82e12T + 1.97e24T^{2} \) |
| 23 | \( 1 + 1.15e13T + 7.46e25T^{2} \) |
| 29 | \( 1 + 8.95e13T + 6.10e27T^{2} \) |
| 31 | \( 1 + 2.42e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 8.35e14T + 6.24e29T^{2} \) |
| 41 | \( 1 - 4.03e14T + 4.39e30T^{2} \) |
| 43 | \( 1 + 2.07e15T + 1.08e31T^{2} \) |
| 47 | \( 1 + 1.47e15T + 5.88e31T^{2} \) |
| 53 | \( 1 + 2.12e16T + 5.77e32T^{2} \) |
| 59 | \( 1 - 1.16e17T + 4.42e33T^{2} \) |
| 61 | \( 1 - 5.68e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 2.16e17T + 4.95e34T^{2} \) |
| 71 | \( 1 - 1.24e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 3.28e17T + 2.53e35T^{2} \) |
| 79 | \( 1 + 1.32e18T + 1.13e36T^{2} \) |
| 83 | \( 1 + 2.45e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 2.81e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 7.51e18T + 5.60e37T^{2} \) |
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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
−9.795562856829146553207234009297, −9.170737813342117080052560201556, −8.383969718403724750914752421027, −7.09316015352086248507391118359, −6.15421291864494375812637021019, −4.28119900624719379687102625172, −3.58951467635110978216905100102, −2.26054447924433983587128683010, −1.74932908495037946621832452502, 0,
1.74932908495037946621832452502, 2.26054447924433983587128683010, 3.58951467635110978216905100102, 4.28119900624719379687102625172, 6.15421291864494375812637021019, 7.09316015352086248507391118359, 8.383969718403724750914752421027, 9.170737813342117080052560201556, 9.795562856829146553207234009297
