| L(s) = 1 | + 16·7-s − 26·9-s − 28·17-s + 304·23-s − 138·25-s + 448·31-s + 140·41-s + 672·47-s − 494·49-s − 416·63-s + 144·71-s + 588·73-s − 928·79-s − 53·81-s − 532·89-s + 1.98e3·97-s − 2.35e3·103-s − 3.42e3·113-s − 448·119-s − 2.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 728·153-s + ⋯ |
| L(s) = 1 | + 0.863·7-s − 0.962·9-s − 0.399·17-s + 2.75·23-s − 1.10·25-s + 2.59·31-s + 0.533·41-s + 2.08·47-s − 1.44·49-s − 0.831·63-s + 0.240·71-s + 0.942·73-s − 1.32·79-s − 0.0727·81-s − 0.633·89-s + 2.08·97-s − 2.24·103-s − 2.84·113-s − 0.345·119-s − 1.81·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.384·153-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.490871612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.490871612\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 26 T^{2} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 138 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2410 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 1594 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 12346 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 152 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23578 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 42058 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 70 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 33878 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 336 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 296746 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 125130 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 444890 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 571034 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 72 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 294 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 464 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 846522 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 266 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 994 T + p^{3} T^{2} )^{2} \) |
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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
−11.66933170036289674095422955238, −11.42728871367842068017936188095, −10.87046686985733600985108456487, −10.59337854121266809100330881755, −9.879830724625259311823062510702, −9.322421705185165082758585753221, −8.852540282437792563225153878739, −8.488634355589628470945753081801, −7.85881878759523047658420363798, −7.58178639920647970867712128052, −6.57749887798527065367562507660, −6.53563018782053499916508501870, −5.50248578807842579568282605460, −5.23290807731858407769584417162, −4.54468078533690786678526468450, −3.99381147066427059913037887625, −2.84465829878519984602927899742, −2.69995710749898665387943606420, −1.48296755003719991447720728344, −0.66043675447049262693983069160,
0.66043675447049262693983069160, 1.48296755003719991447720728344, 2.69995710749898665387943606420, 2.84465829878519984602927899742, 3.99381147066427059913037887625, 4.54468078533690786678526468450, 5.23290807731858407769584417162, 5.50248578807842579568282605460, 6.53563018782053499916508501870, 6.57749887798527065367562507660, 7.58178639920647970867712128052, 7.85881878759523047658420363798, 8.488634355589628470945753081801, 8.852540282437792563225153878739, 9.322421705185165082758585753221, 9.879830724625259311823062510702, 10.59337854121266809100330881755, 10.87046686985733600985108456487, 11.42728871367842068017936188095, 11.66933170036289674095422955238
