| L(s) = 1 | − 512·2-s − 1.30e4·3-s + 2.62e5·4-s + 6.54e6·5-s + 6.70e6·6-s + 9.66e7·7-s − 1.34e8·8-s − 9.90e8·9-s − 3.35e9·10-s + 1.17e10·11-s − 3.43e9·12-s + 3.44e10·13-s − 4.94e10·14-s − 8.57e10·15-s + 6.87e10·16-s − 4.00e11·17-s + 5.07e11·18-s + 8.14e11·19-s + 1.71e12·20-s − 1.26e12·21-s − 6.04e12·22-s − 4.93e12·23-s + 1.75e12·24-s + 2.37e13·25-s − 1.76e13·26-s + 2.81e13·27-s + 2.53e13·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.384·3-s + 1/2·4-s + 1.49·5-s + 0.271·6-s + 0.905·7-s − 0.353·8-s − 0.852·9-s − 1.05·10-s + 1.50·11-s − 0.192·12-s + 0.899·13-s − 0.640·14-s − 0.575·15-s + 1/4·16-s − 0.819·17-s + 0.602·18-s + 0.579·19-s + 0.749·20-s − 0.347·21-s − 1.06·22-s − 0.571·23-s + 0.135·24-s + 1.24·25-s − 0.636·26-s + 0.711·27-s + 0.452·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+19/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(1.312035080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.312035080\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{9} T \) |
| good | 3 | \( 1 + 4364 p T + p^{19} T^{2} \) |
| 5 | \( 1 - 52374 p^{3} T + p^{19} T^{2} \) |
| 7 | \( 1 - 96674264 T + p^{19} T^{2} \) |
| 11 | \( 1 - 11799694452 T + p^{19} T^{2} \) |
| 13 | \( 1 - 2646286766 p T + p^{19} T^{2} \) |
| 17 | \( 1 + 23570447598 p T + p^{19} T^{2} \) |
| 19 | \( 1 - 814875924620 T + p^{19} T^{2} \) |
| 23 | \( 1 + 4937767258872 T + p^{19} T^{2} \) |
| 29 | \( 1 + 96707212093050 T + p^{19} T^{2} \) |
| 31 | \( 1 + 58447954952608 T + p^{19} T^{2} \) |
| 37 | \( 1 - 246079341597854 T + p^{19} T^{2} \) |
| 41 | \( 1 + 2049265663743558 T + p^{19} T^{2} \) |
| 43 | \( 1 - 5698694101737428 T + p^{19} T^{2} \) |
| 47 | \( 1 + 241487233520496 T + p^{19} T^{2} \) |
| 53 | \( 1 + 16046376246286002 T + p^{19} T^{2} \) |
| 59 | \( 1 + 93238940947295100 T + p^{19} T^{2} \) |
| 61 | \( 1 + 41317614065038618 T + p^{19} T^{2} \) |
| 67 | \( 1 - 98205550162519964 T + p^{19} T^{2} \) |
| 71 | \( 1 + 104472325601031528 T + p^{19} T^{2} \) |
| 73 | \( 1 + 171327195230673382 T + p^{19} T^{2} \) |
| 79 | \( 1 + 1498327037960173840 T + p^{19} T^{2} \) |
| 83 | \( 1 - 311954564984060748 T + p^{19} T^{2} \) |
| 89 | \( 1 - 1106996465738312010 T + p^{19} T^{2} \) |
| 97 | \( 1 + 11800957746149561566 T + p^{19} T^{2} \) |
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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
−24.57417224366458349927386855492, −22.10964121529051679758370773498, −20.41705539767213077963378096591, −17.95973734111144147904455991616, −17.02733753345267717043357454376, −14.16129590953618298868502678580, −11.24120446306495439271389313271, −9.083581288116968962779107853524, −6.03429904126089929688409579929, −1.59394266659421640919546674373,
1.59394266659421640919546674373, 6.03429904126089929688409579929, 9.083581288116968962779107853524, 11.24120446306495439271389313271, 14.16129590953618298868502678580, 17.02733753345267717043357454376, 17.95973734111144147904455991616, 20.41705539767213077963378096591, 22.10964121529051679758370773498, 24.57417224366458349927386855492
