| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s + 3·9-s − 12·11-s − 2·12-s + 16-s − 3·18-s + 12·22-s + 2·24-s − 6·25-s − 4·27-s − 32-s + 24·33-s + 3·36-s + 20·41-s − 24·43-s − 12·44-s − 2·48-s − 10·49-s + 6·50-s + 4·54-s − 24·59-s + 64-s − 24·66-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.353·8-s + 9-s − 3.61·11-s − 0.577·12-s + 1/4·16-s − 0.707·18-s + 2.55·22-s + 0.408·24-s − 6/5·25-s − 0.769·27-s − 0.176·32-s + 4.17·33-s + 1/2·36-s + 3.12·41-s − 3.65·43-s − 1.80·44-s − 0.288·48-s − 1.42·49-s + 0.848·50-s + 0.544·54-s − 3.12·59-s + 1/8·64-s − 2.95·66-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 609408 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 609408 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
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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
−7.72047880217154070655817213438, −7.52450739343745495455790166819, −7.48954786032446015166944686975, −6.30687778302223289318527264405, −6.25250569926844252670184221360, −5.66537004424004383201735558506, −5.23091445179449772114465442965, −4.78570582427944795062347753374, −4.45808410287090706931387677434, −3.30037715777891240894020217870, −2.91631229938310047782795220333, −2.23440726888925536869693203702, −1.51685500861613377868289358777, 0, 0,
1.51685500861613377868289358777, 2.23440726888925536869693203702, 2.91631229938310047782795220333, 3.30037715777891240894020217870, 4.45808410287090706931387677434, 4.78570582427944795062347753374, 5.23091445179449772114465442965, 5.66537004424004383201735558506, 6.25250569926844252670184221360, 6.30687778302223289318527264405, 7.48954786032446015166944686975, 7.52450739343745495455790166819, 7.72047880217154070655817213438
