| L(s) = 1 | + 1.63e4·2-s + 6.72e6·3-s + 2.68e8·4-s + 6.10e9·5-s + 1.10e11·6-s + 8.27e11·7-s + 4.39e12·8-s + 2.23e13·9-s + 1.00e14·10-s + 1.80e15·12-s + 1.35e16·14-s + 4.10e16·15-s + 7.20e16·16-s + 3.65e17·18-s + 1.63e18·20-s + 5.56e18·21-s − 1.24e19·23-s + 2.95e19·24-s + 3.72e19·25-s − 3.69e18·27-s + 2.22e20·28-s − 3.98e20·29-s + 6.72e20·30-s + 1.18e21·32-s + 5.04e21·35-s + 5.99e21·36-s + 2.68e22·40-s + ⋯ |
| L(s) = 1 | + 2-s + 1.40·3-s + 4-s + 5-s + 1.40·6-s + 1.21·7-s + 8-s + 0.975·9-s + 10-s + 1.40·12-s + 1.21·14-s + 1.40·15-s + 16-s + 0.975·18-s + 20-s + 1.71·21-s − 1.07·23-s + 1.40·24-s + 25-s − 0.0337·27-s + 1.21·28-s − 1.34·29-s + 1.40·30-s + 32-s + 1.21·35-s + 0.975·36-s + 40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(29-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+14) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{29}{2})\) |
\(\approx\) |
\(10.50714490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.50714490\) |
| \(L(15)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{14} T \) |
| 5 | \( 1 - p^{14} T \) |
| good | 3 | \( 1 - 6723358 T + p^{28} T^{2} \) |
| 7 | \( 1 - 827222074478 T + p^{28} T^{2} \) |
| 11 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 13 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 17 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 19 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 23 | \( 1 + 12490410244449170002 T + p^{28} T^{2} \) |
| 29 | \( 1 + \)\(39\!\cdots\!58\)\( T + p^{28} T^{2} \) |
| 31 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 37 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 41 | \( 1 + \)\(64\!\cdots\!98\)\( T + p^{28} T^{2} \) |
| 43 | \( 1 - \)\(12\!\cdots\!18\)\( T + p^{28} T^{2} \) |
| 47 | \( 1 + \)\(42\!\cdots\!42\)\( T + p^{28} T^{2} \) |
| 53 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 59 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 61 | \( 1 + \)\(15\!\cdots\!98\)\( T + p^{28} T^{2} \) |
| 67 | \( 1 - \)\(36\!\cdots\!58\)\( T + p^{28} T^{2} \) |
| 71 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 73 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 79 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
| 83 | \( 1 + \)\(13\!\cdots\!62\)\( T + p^{28} T^{2} \) |
| 89 | \( 1 + \)\(36\!\cdots\!98\)\( T + p^{28} T^{2} \) |
| 97 | \( ( 1 - p^{14} T )( 1 + p^{14} T ) \) |
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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
−12.86572219081296939187020212628, −11.30315120027015382455648541961, −9.907813866205140943398825586019, −8.494127224705859993652275396982, −7.42530368286348081567521498130, −5.84220748495519752354420429438, −4.61961196781501139967430864873, −3.33917847862162842559781673786, −2.12891709555559780628519583778, −1.62310464442896221557613274750,
1.62310464442896221557613274750, 2.12891709555559780628519583778, 3.33917847862162842559781673786, 4.61961196781501139967430864873, 5.84220748495519752354420429438, 7.42530368286348081567521498130, 8.494127224705859993652275396982, 9.907813866205140943398825586019, 11.30315120027015382455648541961, 12.86572219081296939187020212628
