| L(s) = 1 | + 4·3-s − 4·5-s + 6·9-s − 2·11-s − 16·15-s − 8·23-s + 2·25-s − 4·27-s − 8·33-s − 20·37-s − 24·45-s + 16·47-s + 2·49-s + 12·53-s + 8·55-s + 28·59-s + 20·67-s − 32·69-s − 24·71-s + 8·75-s − 37·81-s − 4·89-s − 4·97-s − 12·99-s + 8·103-s − 80·111-s + 4·113-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s + 2·9-s − 0.603·11-s − 4.13·15-s − 1.66·23-s + 2/5·25-s − 0.769·27-s − 1.39·33-s − 3.28·37-s − 3.57·45-s + 2.33·47-s + 2/7·49-s + 1.64·53-s + 1.07·55-s + 3.64·59-s + 2.44·67-s − 3.85·69-s − 2.84·71-s + 0.923·75-s − 4.11·81-s − 0.423·89-s − 0.406·97-s − 1.20·99-s + 0.788·103-s − 7.59·111-s + 0.376·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.671818297\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.671818297\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.051397256544459587474063215822, −7.33738117792255002879146989083, −7.28624614008190284194684658876, −6.95203079036524006513160995367, −5.94748687441979739999249646343, −5.57971042236015603393981708123, −5.16542238567969836254694300670, −4.22499492266902913993361093129, −3.91058908136761919842529082491, −3.74391345354838938964523130661, −3.39209478584729487520505941993, −2.63262233065457230393893973638, −2.30676446591740616355718333113, −1.82152115990232001116828319877, −0.42520644735956268098029720198,
0.42520644735956268098029720198, 1.82152115990232001116828319877, 2.30676446591740616355718333113, 2.63262233065457230393893973638, 3.39209478584729487520505941993, 3.74391345354838938964523130661, 3.91058908136761919842529082491, 4.22499492266902913993361093129, 5.16542238567969836254694300670, 5.57971042236015603393981708123, 5.94748687441979739999249646343, 6.95203079036524006513160995367, 7.28624614008190284194684658876, 7.33738117792255002879146989083, 8.051397256544459587474063215822
