| L(s) = 1 | + 2·5-s + 9-s − 4·13-s + 4·17-s + 3·25-s − 4·29-s − 4·37-s + 4·41-s + 2·45-s + 49-s − 20·53-s + 12·61-s − 8·65-s − 12·73-s + 81-s + 8·85-s + 4·89-s − 28·97-s − 20·101-s − 4·109-s − 12·113-s − 4·117-s − 6·121-s + 4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1/3·9-s − 1.10·13-s + 0.970·17-s + 3/5·25-s − 0.742·29-s − 0.657·37-s + 0.624·41-s + 0.298·45-s + 1/7·49-s − 2.74·53-s + 1.53·61-s − 0.992·65-s − 1.40·73-s + 1/9·81-s + 0.867·85-s + 0.423·89-s − 2.84·97-s − 1.99·101-s − 0.383·109-s − 1.12·113-s − 0.369·117-s − 0.545·121-s + 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1411200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1411200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−7.63826170331776401843786393867, −7.38480877575677725511614353546, −6.78615026121627054957884200804, −6.49541389816732821186277653939, −5.92792209104066052839413021172, −5.48998134657804488643602211945, −5.10751363306408757224852565788, −4.77131487081713309676703024712, −4.01092057725677360635799148336, −3.66798236875966089815086725641, −2.70612473469008799964811139117, −2.68877128402395436264166966828, −1.69445881682715379624957094269, −1.31138766611309020615751361021, 0,
1.31138766611309020615751361021, 1.69445881682715379624957094269, 2.68877128402395436264166966828, 2.70612473469008799964811139117, 3.66798236875966089815086725641, 4.01092057725677360635799148336, 4.77131487081713309676703024712, 5.10751363306408757224852565788, 5.48998134657804488643602211945, 5.92792209104066052839413021172, 6.49541389816732821186277653939, 6.78615026121627054957884200804, 7.38480877575677725511614353546, 7.63826170331776401843786393867
