| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 3·9-s + 4·11-s + 2·12-s + 16-s − 4·17-s + 3·18-s + 4·22-s + 2·24-s + 4·27-s + 32-s + 8·33-s − 4·34-s + 3·36-s + 4·41-s + 8·43-s + 4·44-s + 2·48-s − 10·49-s − 8·51-s + 4·54-s + 20·59-s + 64-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 9-s + 1.20·11-s + 0.577·12-s + 1/4·16-s − 0.970·17-s + 0.707·18-s + 0.852·22-s + 0.408·24-s + 0.769·27-s + 0.176·32-s + 1.39·33-s − 0.685·34-s + 1/2·36-s + 0.624·41-s + 1.21·43-s + 0.603·44-s + 0.288·48-s − 1.42·49-s − 1.12·51-s + 0.544·54-s + 2.60·59-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.568445638\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.568445638\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 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
−8.294621404044321153232941656353, −7.82916843904617298969738068421, −7.47947512213883760565427163251, −6.73848316801603166260924120780, −6.64959962043719359186062300669, −6.25629768791183300682565415383, −5.34862637188603959556918580152, −5.13544146251726925083565635926, −4.30363492128197640914363500647, −3.90195979683780024724235112717, −3.79143046769365773246334925917, −2.87323565715695326749103005451, −2.45815773509881651106722995929, −1.85419289855368654610516185561, −1.03766093095475251256503366567,
1.03766093095475251256503366567, 1.85419289855368654610516185561, 2.45815773509881651106722995929, 2.87323565715695326749103005451, 3.79143046769365773246334925917, 3.90195979683780024724235112717, 4.30363492128197640914363500647, 5.13544146251726925083565635926, 5.34862637188603959556918580152, 6.25629768791183300682565415383, 6.64959962043719359186062300669, 6.73848316801603166260924120780, 7.47947512213883760565427163251, 7.82916843904617298969738068421, 8.294621404044321153232941656353
