| L(s) = 1 | + 2·2-s + 4·3-s + 2·4-s + 8·6-s + 8·9-s − 12·11-s + 8·12-s − 4·16-s + 8·17-s + 16·18-s − 24·22-s + 12·27-s − 8·32-s − 48·33-s + 16·34-s + 16·36-s − 12·41-s + 12·43-s − 24·44-s − 16·48-s + 32·51-s + 24·54-s − 8·64-s − 96·66-s − 12·67-s + 16·68-s + 24·73-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 2.30·3-s + 4-s + 3.26·6-s + 8/3·9-s − 3.61·11-s + 2.30·12-s − 16-s + 1.94·17-s + 3.77·18-s − 5.11·22-s + 2.30·27-s − 1.41·32-s − 8.35·33-s + 2.74·34-s + 8/3·36-s − 1.87·41-s + 1.82·43-s − 3.61·44-s − 2.30·48-s + 4.48·51-s + 3.26·54-s − 64-s − 11.8·66-s − 1.46·67-s + 1.94·68-s + 2.80·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.540107746\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.540107746\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 146 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
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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
−12.83461125882297384840720534982, −12.59632793749163586086107653210, −12.16709337580430776584670510375, −11.31279950412465471742841070728, −10.59309616654074349556559346898, −10.20659584223364561574361795293, −9.906132805346180439449932564624, −9.024174749586312479583394624138, −8.684520954755740637860495321363, −7.940817487311768164169907409616, −7.67781266462110170830955028344, −7.56544907154702072696127104846, −6.43392652598721456379763884018, −5.53793135770770986041082780606, −5.20188861543177951230871158341, −4.67995772602138674958764101542, −3.44399200216977935047946699399, −3.38425863832263344277539571706, −2.53944393185318725803307799768, −2.36282794789362496136437965985,
2.36282794789362496136437965985, 2.53944393185318725803307799768, 3.38425863832263344277539571706, 3.44399200216977935047946699399, 4.67995772602138674958764101542, 5.20188861543177951230871158341, 5.53793135770770986041082780606, 6.43392652598721456379763884018, 7.56544907154702072696127104846, 7.67781266462110170830955028344, 7.940817487311768164169907409616, 8.684520954755740637860495321363, 9.024174749586312479583394624138, 9.906132805346180439449932564624, 10.20659584223364561574361795293, 10.59309616654074349556559346898, 11.31279950412465471742841070728, 12.16709337580430776584670510375, 12.59632793749163586086107653210, 12.83461125882297384840720534982
