| L(s) = 1 | − 2-s + 4-s − 8-s + 6·11-s + 16-s + 6·17-s − 2·19-s − 6·22-s − 10·25-s − 32-s − 6·34-s + 2·38-s − 18·41-s − 2·43-s + 6·44-s − 10·49-s + 10·50-s − 6·59-s + 64-s + 10·67-s + 6·68-s + 22·73-s − 2·76-s + 18·82-s − 24·83-s + 2·86-s − 6·88-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s + 1.80·11-s + 1/4·16-s + 1.45·17-s − 0.458·19-s − 1.27·22-s − 2·25-s − 0.176·32-s − 1.02·34-s + 0.324·38-s − 2.81·41-s − 0.304·43-s + 0.904·44-s − 1.42·49-s + 1.41·50-s − 0.781·59-s + 1/8·64-s + 1.22·67-s + 0.727·68-s + 2.57·73-s − 0.229·76-s + 1.98·82-s − 2.63·83-s + 0.215·86-s − 0.639·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 839808 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 839808 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−7.930769578371338402756573985837, −7.80670854475898716623947697687, −7.03958314562926506761932301326, −6.68619713403218599725876916726, −6.38621009962525321192846394616, −5.86722110566278992511976681176, −5.34383976963704470462884839275, −4.86353492334106993174795804756, −4.05064804668395326172520142038, −3.65076060485342096132488694617, −3.35055085317237919419053143292, −2.42466756648709547485696737515, −1.60693251106342657124988453680, −1.34471597583379685537750572425, 0,
1.34471597583379685537750572425, 1.60693251106342657124988453680, 2.42466756648709547485696737515, 3.35055085317237919419053143292, 3.65076060485342096132488694617, 4.05064804668395326172520142038, 4.86353492334106993174795804756, 5.34383976963704470462884839275, 5.86722110566278992511976681176, 6.38621009962525321192846394616, 6.68619713403218599725876916726, 7.03958314562926506761932301326, 7.80670854475898716623947697687, 7.930769578371338402756573985837
