| L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·6-s − 8-s − 3·9-s − 2·12-s + 16-s − 14·17-s + 3·18-s − 2·19-s + 2·24-s + 25-s + 14·27-s − 32-s + 14·34-s − 3·36-s + 2·38-s + 4·41-s + 12·43-s − 2·48-s − 13·49-s − 50-s + 28·51-s − 14·54-s + 4·57-s − 14·59-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 − 0.577·12-s + 1/4·16-s − 3.39·17-s + 0.707·18-s − 0.458·19-s + 0.408·24-s + 1/5·25-s + 2.69·27-s − 0.176·32-s + 2.40·34-s − 1/2·36-s + 0.324·38-s + 0.624·41-s + 1.82·43-s − 0.288·48-s − 1.85·49-s − 0.141·50-s + 3.92·51-s − 1.90·54-s + 0.529·57-s − 1.82·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155200 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 18 T + p T^{2} )^{2} \) |
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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
−7.70839715805139448605540433788, −6.88551821385603277867593120567, −6.74268114763584697011939134634, −6.36025544969418794968718402286, −5.93621501809148728100755282492, −5.56345185004455097222791799116, −4.94709938868688158912695317221, −4.42087197415110762692909845352, −4.19480390705796823353792434761, −3.15063118485916249534122395283, −2.58032194053261503171562760419, −2.27566375882933227798874081338, −1.29040153699271294413682322178, 0, 0,
1.29040153699271294413682322178, 2.27566375882933227798874081338, 2.58032194053261503171562760419, 3.15063118485916249534122395283, 4.19480390705796823353792434761, 4.42087197415110762692909845352, 4.94709938868688158912695317221, 5.56345185004455097222791799116, 5.93621501809148728100755282492, 6.36025544969418794968718402286, 6.74268114763584697011939134634, 6.88551821385603277867593120567, 7.70839715805139448605540433788
