L(s) = 1 | + 2·3-s + 3·9-s − 6·11-s − 6·17-s − 2·19-s + 10·27-s − 12·33-s − 6·41-s + 20·43-s − 14·49-s − 12·51-s − 4·57-s + 12·59-s − 14·67-s − 2·73-s + 20·81-s + 18·83-s + 18·89-s − 20·97-s − 18·99-s + 6·107-s + 18·113-s + 11·121-s − 12·123-s + 127-s + 40·129-s + 131-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 9-s − 1.80·11-s − 1.45·17-s − 0.458·19-s + 1.92·27-s − 2.08·33-s − 0.937·41-s + 3.04·43-s − 2·49-s − 1.68·51-s − 0.529·57-s + 1.56·59-s − 1.71·67-s − 0.234·73-s + 20/9·81-s + 1.97·83-s + 1.90·89-s − 2.03·97-s − 1.80·99-s + 0.580·107-s + 1.69·113-s + 121-s − 1.08·123-s + 0.0887·127-s + 3.52·129-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.537997386\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.537997386\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 - 2 T + T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 18 T + 241 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−8.158269619328284288121111761745, −7.981406316474967923561262047143, −7.54104063044905995247567927639, −7.10078339181942506521685236508, −7.09951976988719118660533802501, −6.41258367784671057717372410803, −6.14077178094832556374098490376, −5.84090041567008113679222673009, −5.07301536728739105917829876294, −5.00044092370209343587505490124, −4.59681357344296214218642119884, −4.23018575494668309822152590783, −3.76843093203355239813953280493, −3.31194506747801410241492322345, −2.83708893944802766926779860041, −2.60064035993402745107253429976, −2.14872520717401056183623429748, −1.89443979378563223026318462878, −1.05267532629939590992481923679, −0.38220343061120510809769277337,
0.38220343061120510809769277337, 1.05267532629939590992481923679, 1.89443979378563223026318462878, 2.14872520717401056183623429748, 2.60064035993402745107253429976, 2.83708893944802766926779860041, 3.31194506747801410241492322345, 3.76843093203355239813953280493, 4.23018575494668309822152590783, 4.59681357344296214218642119884, 5.00044092370209343587505490124, 5.07301536728739105917829876294, 5.84090041567008113679222673009, 6.14077178094832556374098490376, 6.41258367784671057717372410803, 7.09951976988719118660533802501, 7.10078339181942506521685236508, 7.54104063044905995247567927639, 7.981406316474967923561262047143, 8.158269619328284288121111761745