| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 3·8-s + 3·9-s + 4·11-s + 2·12-s − 16-s − 10·17-s − 3·18-s − 4·19-s − 4·22-s − 6·24-s + 4·25-s − 4·27-s − 5·32-s − 8·33-s + 10·34-s − 3·36-s + 4·38-s + 2·41-s − 4·44-s + 2·48-s − 10·49-s − 4·50-s + 20·51-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.06·8-s + 9-s + 1.20·11-s + 0.577·12-s − 1/4·16-s − 2.42·17-s − 0.707·18-s − 0.917·19-s − 0.852·22-s − 1.22·24-s + 4/5·25-s − 0.769·27-s − 0.883·32-s − 1.39·33-s + 1.71·34-s − 1/2·36-s + 0.648·38-s + 0.312·41-s − 0.603·44-s + 0.288·48-s − 1.42·49-s − 0.565·50-s + 2.80·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65088 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65088 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 113 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 18 T + p T^{2} ) \) |
| good | 5 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
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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
−9.432991146143495649740576290596, −9.280312475303777417166371019214, −8.731972829539004946056948269295, −8.293230044260983737438432122323, −7.59788600191101188963379174899, −6.84919786059842385556217497980, −6.57836641693906311829837488696, −6.19862920953773918938154386596, −5.26593518456827044450952011536, −4.64674904149900113268310841641, −4.35254653967126239963919199666, −3.69100845554721173847989496284, −2.27389651117292666071654437143, −1.33455359339994372982372277656, 0,
1.33455359339994372982372277656, 2.27389651117292666071654437143, 3.69100845554721173847989496284, 4.35254653967126239963919199666, 4.64674904149900113268310841641, 5.26593518456827044450952011536, 6.19862920953773918938154386596, 6.57836641693906311829837488696, 6.84919786059842385556217497980, 7.59788600191101188963379174899, 8.293230044260983737438432122323, 8.731972829539004946056948269295, 9.280312475303777417166371019214, 9.432991146143495649740576290596
