| L(s) = 1 | + 1.02e3·2-s + 5.24e5·4-s + 7.13e6·5-s + 7.30e9·10-s + 6.18e10·13-s − 2.74e11·16-s − 1.03e12·17-s + 3.73e12·20-s + 3.17e13·25-s + 6.33e13·26-s − 2.81e14·32-s − 1.05e15·34-s + 1.58e15·37-s + 2.10e15·41-s + 3.25e16·50-s + 3.24e16·52-s + 6.64e16·53-s + 2.94e17·61-s − 1.44e17·64-s + 4.41e17·65-s − 5.41e17·68-s − 3.54e17·73-s + 1.62e18·74-s − 1.96e18·80-s − 1.35e18·81-s + 2.15e18·82-s − 7.36e18·85-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.63·5-s + 2.30·10-s + 1.61·13-s − 16-s − 2.11·17-s + 1.63·20-s + 1.66·25-s + 2.28·26-s − 1.41·32-s − 2.98·34-s + 2.00·37-s + 1.00·41-s + 2.35·50-s + 1.61·52-s + 2.76·53-s + 3.22·61-s − 64-s + 2.64·65-s − 2.11·68-s − 0.705·73-s + 2.83·74-s − 1.63·80-s − 81-s + 1.41·82-s − 3.44·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(20-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+19/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(10)\) |
\(\approx\) |
\(9.853236952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.853236952\) |
| \(L(\frac{21}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 - p^{10} T + p^{19} T^{2} \) |
| 5 | $C_2$ | \( 1 - 7131836 T + p^{19} T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + p^{38} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{38} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 75282446308 T + p^{19} T^{2} )( 1 + 13425142062 T + p^{19} T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 56485706776 T + p^{19} T^{2} )( 1 + 976267948706 T + p^{19} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{38} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 91338489869890 T + p^{19} T^{2} )( 1 + 91338489869890 T + p^{19} T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 1581041579948724 T + p^{19} T^{2} )( 1 - 5957049320854 T + p^{19} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 1052302186804312 T + p^{19} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{38} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{38} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 40311272635409668 T + p^{19} T^{2} )( 1 - 26140974121322438 T + p^{19} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 147206587324435692 T + p^{19} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{38} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p^{19} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 511563528750872778 T + p^{19} T^{2} )( 1 + 866218773026938192 T + p^{19} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{19} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{38} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6133570290214295290 T + p^{19} T^{2} )( 1 + 6133570290214295290 T + p^{19} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 1379924894825164206 T + p^{19} T^{2} )( 1 + 14911099796840662936 T + p^{19} T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.77458365991368082597858049888, −13.70037814339460556057423467516, −12.94752262909603378138486282600, −12.92380775609969796463479800014, −11.57411979008036015361952840962, −11.17033916111199841236586282372, −10.47806596311580955665628610399, −9.536720790533272848374031165133, −8.970156280804823178796724534495, −8.353942666323463848724004902092, −6.74132301989927563092154402920, −6.62069418425032971080860231793, −5.61806225330882180027996075685, −5.54769879512638355917145413623, −4.21843244928944180199271520985, −4.05534791362563443536232101144, −2.64189942041799017984812574120, −2.45685943899910210225768022382, −1.52323835851904231884095491935, −0.66578520317204816501685122401,
0.66578520317204816501685122401, 1.52323835851904231884095491935, 2.45685943899910210225768022382, 2.64189942041799017984812574120, 4.05534791362563443536232101144, 4.21843244928944180199271520985, 5.54769879512638355917145413623, 5.61806225330882180027996075685, 6.62069418425032971080860231793, 6.74132301989927563092154402920, 8.353942666323463848724004902092, 8.970156280804823178796724534495, 9.536720790533272848374031165133, 10.47806596311580955665628610399, 11.17033916111199841236586282372, 11.57411979008036015361952840962, 12.92380775609969796463479800014, 12.94752262909603378138486282600, 13.70037814339460556057423467516, 13.77458365991368082597858049888
