| L(s) = 1 | − 4-s + 2·9-s + 16-s + 4·19-s + 12·29-s + 8·31-s − 2·36-s − 12·41-s − 12·59-s − 16·61-s − 64-s − 4·76-s − 16·79-s − 5·81-s − 12·89-s − 4·109-s − 12·116-s − 22·121-s − 8·124-s + 127-s + 131-s + 137-s + 139-s + 2·144-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2/3·9-s + 1/4·16-s + 0.917·19-s + 2.22·29-s + 1.43·31-s − 1/3·36-s − 1.87·41-s − 1.56·59-s − 2.04·61-s − 1/8·64-s − 0.458·76-s − 1.80·79-s − 5/9·81-s − 1.27·89-s − 0.383·109-s − 1.11·116-s − 2·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1/6·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.018939629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.018939629\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 94 T^{2} + p^{2} T^{4} \) |
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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
−9.291580408060141983979399186092, −8.686799213988998412180612357826, −8.372317926425118838681390711910, −7.974685124744191518425915416230, −7.77997128109805632868382916423, −7.07256928059767764542463318469, −6.94757629042622922564060045741, −6.30169340373003154799973259917, −6.26241248410960786189616124972, −5.41015405814042970595648062909, −5.29080240231530046836053673283, −4.52680662177885794476002978576, −4.52338079867262475696931665972, −4.08804702133336435501210645596, −3.17540836021014981116902664595, −3.10579981581082720866300495059, −2.60263928118040302973129514711, −1.54539346394132368876531348887, −1.39715208919112226567774463728, −0.51355078800606068609580219707,
0.51355078800606068609580219707, 1.39715208919112226567774463728, 1.54539346394132368876531348887, 2.60263928118040302973129514711, 3.10579981581082720866300495059, 3.17540836021014981116902664595, 4.08804702133336435501210645596, 4.52338079867262475696931665972, 4.52680662177885794476002978576, 5.29080240231530046836053673283, 5.41015405814042970595648062909, 6.26241248410960786189616124972, 6.30169340373003154799973259917, 6.94757629042622922564060045741, 7.07256928059767764542463318469, 7.77997128109805632868382916423, 7.974685124744191518425915416230, 8.372317926425118838681390711910, 8.686799213988998412180612357826, 9.291580408060141983979399186092
