| L(s) = 1 | − 5.24e5·2-s − 2.23e9·3-s + 2.74e11·4-s − 1.90e13·5-s + 1.17e15·6-s − 1.59e16·7-s − 1.44e17·8-s + 3.66e18·9-s + 1.00e19·10-s − 6.15e20·12-s + 8.38e21·14-s + 4.27e22·15-s + 7.55e22·16-s − 1.91e24·18-s − 5.24e24·20-s + 3.58e25·21-s + 1.17e26·23-s + 3.22e26·24-s + 3.63e26·25-s − 5.17e27·27-s − 4.39e27·28-s + 1.09e28·29-s − 2.23e28·30-s − 3.96e28·32-s + 3.05e29·35-s + 1.00e30·36-s + 2.74e30·40-s + ⋯ |
| L(s) = 1 | − 2-s − 1.92·3-s + 4-s − 5-s + 1.92·6-s − 1.40·7-s − 8-s + 2.71·9-s + 10-s − 1.92·12-s + 1.40·14-s + 1.92·15-s + 16-s − 2.71·18-s − 20-s + 2.70·21-s + 1.58·23-s + 1.92·24-s + 25-s − 3.29·27-s − 1.40·28-s + 1.79·29-s − 1.92·30-s − 32-s + 1.40·35-s + 2.71·36-s + 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(39-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+19) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{39}{2})\) |
\(\approx\) |
\(0.4750674870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4750674870\) |
| \(L(20)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{19} T \) |
| 5 | \( 1 + p^{19} T \) |
| good | 3 | \( 1 + 2238849644 T + p^{38} T^{2} \) |
| 7 | \( 1 + 15999194597449396 T + p^{38} T^{2} \) |
| 11 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 13 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 17 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 19 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 23 | \( 1 - \)\(11\!\cdots\!36\)\( T + p^{38} T^{2} \) |
| 29 | \( 1 - \)\(10\!\cdots\!58\)\( T + p^{38} T^{2} \) |
| 31 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 37 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 41 | \( 1 - \)\(48\!\cdots\!02\)\( T + p^{38} T^{2} \) |
| 43 | \( 1 - \)\(21\!\cdots\!76\)\( T + p^{38} T^{2} \) |
| 47 | \( 1 - \)\(85\!\cdots\!44\)\( T + p^{38} T^{2} \) |
| 53 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 59 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 61 | \( 1 - \)\(19\!\cdots\!02\)\( T + p^{38} T^{2} \) |
| 67 | \( 1 + \)\(85\!\cdots\!56\)\( T + p^{38} T^{2} \) |
| 71 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 73 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 79 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
| 83 | \( 1 - \)\(22\!\cdots\!16\)\( T + p^{38} T^{2} \) |
| 89 | \( 1 + \)\(21\!\cdots\!02\)\( T + p^{38} T^{2} \) |
| 97 | \( ( 1 - p^{19} T )( 1 + p^{19} T ) \) |
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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
−11.05536434605061346414815119838, −10.36524937350389810490315838184, −9.160207860280308845956665254366, −7.38492015751643034875292140984, −6.70215710349397183360496235558, −5.79725185654709699415497518699, −4.35059581003857761163682836556, −2.93522644410615910151038210355, −0.923971514538432392903714996426, −0.54158087382731914233903410190,
0.54158087382731914233903410190, 0.923971514538432392903714996426, 2.93522644410615910151038210355, 4.35059581003857761163682836556, 5.79725185654709699415497518699, 6.70215710349397183360496235558, 7.38492015751643034875292140984, 9.160207860280308845956665254366, 10.36524937350389810490315838184, 11.05536434605061346414815119838
