| L(s) = 1 | − 4·3-s + 6·9-s − 10·11-s − 6·17-s + 2·19-s − 9·25-s + 4·27-s + 40·33-s + 20·41-s − 2·43-s − 5·49-s + 24·51-s − 8·57-s − 12·59-s + 24·67-s + 18·73-s + 36·75-s − 37·81-s + 24·83-s + 24·89-s − 16·97-s − 60·99-s − 4·107-s − 20·113-s + 53·121-s − 80·123-s + 127-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 2·9-s − 3.01·11-s − 1.45·17-s + 0.458·19-s − 9/5·25-s + 0.769·27-s + 6.96·33-s + 3.12·41-s − 0.304·43-s − 5/7·49-s + 3.36·51-s − 1.05·57-s − 1.56·59-s + 2.93·67-s + 2.10·73-s + 4.15·75-s − 4.11·81-s + 2.63·83-s + 2.54·89-s − 1.62·97-s − 6.03·99-s − 0.386·107-s − 1.88·113-s + 4.81·121-s − 7.21·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369664 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
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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
−8.121865683855346250917871835791, −7.982276472778928545712318850832, −7.61880161741027781839162381092, −6.89999375237673531400272497166, −6.35171638937504994192666881666, −6.06650473039374929409046448898, −5.55162300458149510222680140561, −5.18067831877099909482237924161, −4.93085486739421392203871722376, −4.37905347680940090915441446169, −3.51647821375355615052320307776, −2.48496537170117614857619865014, −2.31923248203641562194456416521, −0.67525660115678139253233977173, 0,
0.67525660115678139253233977173, 2.31923248203641562194456416521, 2.48496537170117614857619865014, 3.51647821375355615052320307776, 4.37905347680940090915441446169, 4.93085486739421392203871722376, 5.18067831877099909482237924161, 5.55162300458149510222680140561, 6.06650473039374929409046448898, 6.35171638937504994192666881666, 6.89999375237673531400272497166, 7.61880161741027781839162381092, 7.982276472778928545712318850832, 8.121865683855346250917871835791
