| L(s) = 1 | − 2.62e5·2-s + 6.87e10·4-s − 4.22e12·5-s − 1.80e16·8-s + 1.50e17·9-s + 1.10e18·10-s − 1.52e20·13-s + 4.72e21·16-s − 2.31e22·17-s − 3.93e22·18-s − 2.90e23·20-s + 3.32e24·25-s + 4.00e25·26-s + 1.78e26·29-s − 1.23e27·32-s + 6.06e27·34-s + 1.03e28·36-s + 3.18e28·37-s + 7.61e28·40-s + 1.42e29·41-s − 6.34e29·45-s + 2.65e30·49-s − 8.72e29·50-s − 1.05e31·52-s − 1.80e31·53-s − 4.68e31·58-s + 2.71e32·61-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 1.10·5-s − 8-s + 9-s + 1.10·10-s − 1.35·13-s + 16-s − 1.64·17-s − 18-s − 1.10·20-s + 0.228·25-s + 1.35·26-s + 0.849·29-s − 32-s + 1.64·34-s + 36-s + 1.88·37-s + 1.10·40-s + 1.33·41-s − 1.10·45-s + 49-s − 0.228·50-s − 1.35·52-s − 1.65·53-s − 0.849·58-s + 1.98·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(37-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+18) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{37}{2})\) |
\(\approx\) |
\(0.7619507044\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7619507044\) |
| \(L(19)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{18} T \) |
| good | 3 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 5 | \( 1 + 4228490555534 T + p^{36} T^{2} \) |
| 7 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 11 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 13 | \( 1 + \)\(15\!\cdots\!58\)\( T + p^{36} T^{2} \) |
| 17 | \( 1 + \)\(23\!\cdots\!18\)\( T + p^{36} T^{2} \) |
| 19 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 23 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 29 | \( 1 - \)\(17\!\cdots\!22\)\( T + p^{36} T^{2} \) |
| 31 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 37 | \( 1 - \)\(31\!\cdots\!42\)\( T + p^{36} T^{2} \) |
| 41 | \( 1 - \)\(14\!\cdots\!42\)\( T + p^{36} T^{2} \) |
| 43 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 47 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 53 | \( 1 + \)\(18\!\cdots\!78\)\( T + p^{36} T^{2} \) |
| 59 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 61 | \( 1 - \)\(27\!\cdots\!62\)\( T + p^{36} T^{2} \) |
| 67 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 71 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 73 | \( 1 - \)\(65\!\cdots\!62\)\( T + p^{36} T^{2} \) |
| 79 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 83 | \( ( 1 - p^{18} T )( 1 + p^{18} T ) \) |
| 89 | \( 1 - \)\(75\!\cdots\!62\)\( T + p^{36} T^{2} \) |
| 97 | \( 1 + \)\(91\!\cdots\!78\)\( T + p^{36} T^{2} \) |
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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
−16.09820561280790544090697773179, −15.12305650194599220394093832924, −12.48482994186258713512356307707, −11.13203178788915375573065473491, −9.579601604908208308242131609897, −7.928361989856889895036035093876, −6.85057642540024914521913034636, −4.33119602369333093714594962690, −2.38593711626350460953623530425, −0.60648030974014511743288830420,
0.60648030974014511743288830420, 2.38593711626350460953623530425, 4.33119602369333093714594962690, 6.85057642540024914521913034636, 7.928361989856889895036035093876, 9.579601604908208308242131609897, 11.13203178788915375573065473491, 12.48482994186258713512356307707, 15.12305650194599220394093832924, 16.09820561280790544090697773179
