| L(s) = 1 | − 4·7-s − 5·9-s + 2·11-s + 8·13-s − 4·16-s − 4·17-s − 9·25-s + 6·37-s − 12·43-s + 16·47-s − 2·49-s − 6·53-s + 10·59-s + 20·63-s − 8·77-s + 16·81-s + 30·89-s − 32·91-s − 14·97-s − 10·99-s + 36·107-s + 16·112-s + 18·113-s − 40·117-s + 16·119-s + 3·121-s + 127-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 5/3·9-s + 0.603·11-s + 2.21·13-s − 16-s − 0.970·17-s − 9/5·25-s + 0.986·37-s − 1.82·43-s + 2.33·47-s − 2/7·49-s − 0.824·53-s + 1.30·59-s + 2.51·63-s − 0.911·77-s + 16/9·81-s + 3.17·89-s − 3.35·91-s − 1.42·97-s − 1.00·99-s + 3.48·107-s + 1.51·112-s + 1.69·113-s − 3.69·117-s + 1.46·119-s + 3/11·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 339889 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 339889 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8850922761\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8850922761\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| 53 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
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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
−8.695157772416545494546569220747, −8.603539619290756001226038948684, −7.897483464878131921600241104256, −7.36095752078983936132318716180, −6.49737245338834450626105272813, −6.36261389471308870138602900888, −6.15894340608527249737055637518, −5.63591512751798630708769003164, −4.91419721756313978845072424427, −4.08847387815843813369306541952, −3.68381000464579683591996966028, −3.29168899169749781933616191290, −2.56273686004108922271717946472, −1.86954239377160649257505077121, −0.51542943980575115852272959956,
0.51542943980575115852272959956, 1.86954239377160649257505077121, 2.56273686004108922271717946472, 3.29168899169749781933616191290, 3.68381000464579683591996966028, 4.08847387815843813369306541952, 4.91419721756313978845072424427, 5.63591512751798630708769003164, 6.15894340608527249737055637518, 6.36261389471308870138602900888, 6.49737245338834450626105272813, 7.36095752078983936132318716180, 7.897483464878131921600241104256, 8.603539619290756001226038948684, 8.695157772416545494546569220747
