| L(s) = 1 | + 2·3-s + 4-s − 3·9-s + 2·12-s + 5·13-s + 16-s + 6·17-s + 6·23-s − 10·25-s − 14·27-s + 18·29-s − 3·36-s + 10·39-s + 16·43-s + 2·48-s − 13·49-s + 12·51-s + 5·52-s − 6·53-s − 20·61-s + 64-s + 6·68-s + 12·69-s − 20·75-s − 20·79-s − 4·81-s + 36·87-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1/2·4-s − 9-s + 0.577·12-s + 1.38·13-s + 1/4·16-s + 1.45·17-s + 1.25·23-s − 2·25-s − 2.69·27-s + 3.34·29-s − 1/2·36-s + 1.60·39-s + 2.43·43-s + 0.288·48-s − 1.85·49-s + 1.68·51-s + 0.693·52-s − 0.824·53-s − 2.56·61-s + 1/8·64-s + 0.727·68-s + 1.44·69-s − 2.30·75-s − 2.25·79-s − 4/9·81-s + 3.85·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 244036 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 244036 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.974155647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.974155647\) |
| \(L(\frac{3}{2})\) |
|
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$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.701620915754948218777480745073, −8.684413720393242419277779815238, −7.969576930662872845166803888368, −7.73591843742988285704805588989, −7.35558189285866919427797765822, −6.25338153432601213583646709472, −6.18304856110129897953773330946, −5.81027585652647529822118508915, −5.00748366820747128480894903619, −4.36756058739341503144365761002, −3.55502409354175041648709074357, −2.99432139105097413830284496966, −2.97707225880527315032523526567, −1.96927933577139967417741574766, −1.05236257727154933297058348414,
1.05236257727154933297058348414, 1.96927933577139967417741574766, 2.97707225880527315032523526567, 2.99432139105097413830284496966, 3.55502409354175041648709074357, 4.36756058739341503144365761002, 5.00748366820747128480894903619, 5.81027585652647529822118508915, 6.18304856110129897953773330946, 6.25338153432601213583646709472, 7.35558189285866919427797765822, 7.73591843742988285704805588989, 7.969576930662872845166803888368, 8.684413720393242419277779815238, 8.701620915754948218777480745073
