| L(s) = 1 | + 4·5-s − 7-s − 6·9-s − 8·11-s + 4·13-s + 2·25-s + 16·31-s − 4·35-s − 8·43-s − 24·45-s − 16·47-s + 49-s − 32·55-s − 12·61-s + 6·63-s + 16·65-s − 8·67-s + 8·77-s + 27·81-s − 4·91-s + 48·99-s + 4·101-s − 32·103-s − 24·107-s + 4·113-s − 24·117-s + 26·121-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 0.377·7-s − 2·9-s − 2.41·11-s + 1.10·13-s + 2/5·25-s + 2.87·31-s − 0.676·35-s − 1.21·43-s − 3.57·45-s − 2.33·47-s + 1/7·49-s − 4.31·55-s − 1.53·61-s + 0.755·63-s + 1.98·65-s − 0.977·67-s + 0.911·77-s + 3·81-s − 0.419·91-s + 4.82·99-s + 0.398·101-s − 3.15·103-s − 2.32·107-s + 0.376·113-s − 2.21·117-s + 2.36·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175616 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( 1 + T \) |
| good | 3 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.000772938990870896314957416552, −8.227785865636410870393529644581, −8.188420417684287628984093773695, −7.84054312303988042972335609462, −6.59264336057149769206055975154, −6.42262084813563967264199691637, −5.91740132798549001211232285063, −5.47493819124631815597461836959, −5.22316409435338364257345412913, −4.53211800578457486655010876487, −3.32773603072291404011890923635, −2.79183800612725741835263061431, −2.58182243115007516792385368988, −1.62407385510972851799092712087, 0,
1.62407385510972851799092712087, 2.58182243115007516792385368988, 2.79183800612725741835263061431, 3.32773603072291404011890923635, 4.53211800578457486655010876487, 5.22316409435338364257345412913, 5.47493819124631815597461836959, 5.91740132798549001211232285063, 6.42262084813563967264199691637, 6.59264336057149769206055975154, 7.84054312303988042972335609462, 8.188420417684287628984093773695, 8.227785865636410870393529644581, 9.000772938990870896314957416552
