| L(s) = 1 | + 2·3-s + 9-s + 4·11-s − 8·13-s + 16·23-s + 2·25-s − 4·27-s + 8·33-s − 8·37-s − 16·39-s + 6·49-s + 12·59-s − 8·61-s + 32·69-s + 16·71-s + 20·73-s + 4·75-s − 11·81-s + 12·83-s − 12·97-s + 4·99-s + 12·107-s − 8·109-s − 16·111-s − 8·117-s − 10·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/3·9-s + 1.20·11-s − 2.21·13-s + 3.33·23-s + 2/5·25-s − 0.769·27-s + 1.39·33-s − 1.31·37-s − 2.56·39-s + 6/7·49-s + 1.56·59-s − 1.02·61-s + 3.85·69-s + 1.89·71-s + 2.34·73-s + 0.461·75-s − 1.22·81-s + 1.31·83-s − 1.21·97-s + 0.402·99-s + 1.16·107-s − 0.766·109-s − 1.51·111-s − 0.739·117-s − 0.909·121-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.923913485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.923913485\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 66 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−10.45943174528697459527542872978, −9.953089882344299536716434294638, −9.476957162199833906927255029521, −9.263378320586195366362382753612, −8.819446980705997261546477777318, −8.686909875576544106754432464652, −7.84788323485869656032738681067, −7.62001859279758778663206024957, −7.00130961375351693650600377747, −6.83763238364471673847100831033, −6.42927177558659482632124989552, −5.26960162720519770499206824585, −5.21517854425873512394274344406, −4.76938340074289338914595762334, −3.89946518316382175569935990499, −3.58652627780453022808913128848, −2.77733901584989806329493698843, −2.61303592544926057220157268307, −1.79918803859608130451973173990, −0.835778173840597778927808714701,
0.835778173840597778927808714701, 1.79918803859608130451973173990, 2.61303592544926057220157268307, 2.77733901584989806329493698843, 3.58652627780453022808913128848, 3.89946518316382175569935990499, 4.76938340074289338914595762334, 5.21517854425873512394274344406, 5.26960162720519770499206824585, 6.42927177558659482632124989552, 6.83763238364471673847100831033, 7.00130961375351693650600377747, 7.62001859279758778663206024957, 7.84788323485869656032738681067, 8.686909875576544106754432464652, 8.819446980705997261546477777318, 9.263378320586195366362382753612, 9.476957162199833906927255029521, 9.953089882344299536716434294638, 10.45943174528697459527542872978
