| L(s) = 1 | − 891.·2-s − 6.29e4·3-s + 2.69e5·4-s + 4.92e6·5-s + 5.61e7·6-s + 6.71e7·7-s + 2.26e8·8-s + 2.80e9·9-s − 4.38e9·10-s + 9.30e9·11-s − 1.69e10·12-s − 2.54e10·13-s − 5.98e10·14-s − 3.10e11·15-s − 3.43e11·16-s + 5.11e11·17-s − 2.49e12·18-s − 2.09e11·19-s + 1.32e12·20-s − 4.22e12·21-s − 8.28e12·22-s + 1.57e12·23-s − 1.42e13·24-s + 5.18e12·25-s + 2.26e13·26-s − 1.03e14·27-s + 1.81e13·28-s + ⋯ |
| L(s) = 1 | − 1.23·2-s − 1.84·3-s + 0.514·4-s + 1.12·5-s + 2.27·6-s + 0.629·7-s + 0.597·8-s + 2.41·9-s − 1.38·10-s + 1.18·11-s − 0.949·12-s − 0.665·13-s − 0.774·14-s − 2.08·15-s − 1.24·16-s + 1.04·17-s − 2.96·18-s − 0.148·19-s + 0.580·20-s − 1.16·21-s − 1.46·22-s + 0.182·23-s − 1.10·24-s + 0.271·25-s + 0.819·26-s − 2.60·27-s + 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 47 | \( 1 - 1.11e15T \) |
| good | 2 | \( 1 + 891.T + 5.24e5T^{2} \) |
| 3 | \( 1 + 6.29e4T + 1.16e9T^{2} \) |
| 5 | \( 1 - 4.92e6T + 1.90e13T^{2} \) |
| 7 | \( 1 - 6.71e7T + 1.13e16T^{2} \) |
| 11 | \( 1 - 9.30e9T + 6.11e19T^{2} \) |
| 13 | \( 1 + 2.54e10T + 1.46e21T^{2} \) |
| 17 | \( 1 - 5.11e11T + 2.39e23T^{2} \) |
| 19 | \( 1 + 2.09e11T + 1.97e24T^{2} \) |
| 23 | \( 1 - 1.57e12T + 7.46e25T^{2} \) |
| 29 | \( 1 + 1.51e14T + 6.10e27T^{2} \) |
| 31 | \( 1 - 1.23e14T + 2.16e28T^{2} \) |
| 37 | \( 1 - 1.34e15T + 6.24e29T^{2} \) |
| 41 | \( 1 + 1.43e15T + 4.39e30T^{2} \) |
| 43 | \( 1 + 1.93e15T + 1.08e31T^{2} \) |
| 53 | \( 1 + 3.32e16T + 5.77e32T^{2} \) |
| 59 | \( 1 + 2.53e16T + 4.42e33T^{2} \) |
| 61 | \( 1 - 9.80e16T + 8.34e33T^{2} \) |
| 67 | \( 1 + 4.03e17T + 4.95e34T^{2} \) |
| 71 | \( 1 + 6.08e17T + 1.49e35T^{2} \) |
| 73 | \( 1 - 1.75e17T + 2.53e35T^{2} \) |
| 79 | \( 1 - 4.37e17T + 1.13e36T^{2} \) |
| 83 | \( 1 + 5.43e17T + 2.90e36T^{2} \) |
| 89 | \( 1 - 1.57e18T + 1.09e37T^{2} \) |
| 97 | \( 1 + 7.82e18T + 5.60e37T^{2} \) |
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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
−11.07173150861023352398581586618, −9.996007658021846071190434675350, −9.427403214728734977108411816881, −7.64945550969411729829753594476, −6.50277570660392258026404157116, −5.51062670243815946887568166876, −4.45542966618909289180438483374, −1.70554759416709292188934574019, −1.17096333016757547339614213004, 0,
1.17096333016757547339614213004, 1.70554759416709292188934574019, 4.45542966618909289180438483374, 5.51062670243815946887568166876, 6.50277570660392258026404157116, 7.64945550969411729829753594476, 9.427403214728734977108411816881, 9.996007658021846071190434675350, 11.07173150861023352398581586618
