| L(s) = 1 | − 3·9-s + 6·11-s − 3·17-s + 5·19-s + 2·25-s − 17·41-s + 5·43-s − 8·49-s + 8·59-s − 13·67-s − 14·73-s + 9·81-s + 9·83-s + 89-s − 18·97-s − 18·99-s − 8·107-s − 24·113-s + 9·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 9·153-s + 157-s + ⋯ |
| L(s) = 1 | − 9-s + 1.80·11-s − 0.727·17-s + 1.14·19-s + 2/5·25-s − 2.65·41-s + 0.762·43-s − 8/7·49-s + 1.04·59-s − 1.58·67-s − 1.63·73-s + 81-s + 0.987·83-s + 0.105·89-s − 1.82·97-s − 1.80·99-s − 0.773·107-s − 2.25·113-s + 9/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.727·153-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331712 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 17 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 24 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 131 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
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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
−7.85230633966006796404025398236, −7.15758741811198390223974129170, −6.83370510301691614082835539749, −6.56026162937724950267572356625, −5.99267087202173280471535940029, −5.64981632788892622516696634359, −4.98914149416235335219451536453, −4.74859736488083006510501884481, −3.93541891434438593703303431892, −3.68206459341817329010729748693, −3.04793409085581096637883901885, −2.59746762026882187511633433361, −1.66891552653535809808405483051, −1.21556832580894178017149313292, 0,
1.21556832580894178017149313292, 1.66891552653535809808405483051, 2.59746762026882187511633433361, 3.04793409085581096637883901885, 3.68206459341817329010729748693, 3.93541891434438593703303431892, 4.74859736488083006510501884481, 4.98914149416235335219451536453, 5.64981632788892622516696634359, 5.99267087202173280471535940029, 6.56026162937724950267572356625, 6.83370510301691614082835539749, 7.15758741811198390223974129170, 7.85230633966006796404025398236
