| L(s) = 1 | − 4·5-s + 5·9-s + 6·11-s + 11·25-s + 10·29-s − 4·31-s + 4·41-s − 20·45-s − 49-s − 24·55-s + 20·59-s − 16·61-s + 16·71-s − 10·79-s + 16·81-s + 30·99-s + 24·101-s − 10·109-s + 5·121-s − 24·125-s + 127-s + 131-s + 137-s + 139-s − 40·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 5/3·9-s + 1.80·11-s + 11/5·25-s + 1.85·29-s − 0.718·31-s + 0.624·41-s − 2.98·45-s − 1/7·49-s − 3.23·55-s + 2.60·59-s − 2.04·61-s + 1.89·71-s − 1.12·79-s + 16/9·81-s + 3.01·99-s + 2.38·101-s − 0.957·109-s + 5/11·121-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 3.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 313600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.722027016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.722027016\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 145 T^{2} + p^{2} T^{4} \) |
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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
−10.91070189481907922956927318156, −10.71009106781184143203315927908, −10.03091677087469056227918960742, −9.766656387781431151823788433392, −9.129171517005568276859253511243, −8.763191837037744281387501238879, −8.375848115054907090016583855751, −7.70808153712614068374976575634, −7.45311496359253204808358138568, −7.00666108758363224630411901529, −6.46838322567371333364017657592, −6.35389253844449468960839880985, −5.17906672504472111908101207858, −4.74636824709949268335118783326, −4.13719132333362772741355965576, −3.94513573682298272413934743563, −3.50828238529535548574726346732, −2.61115291801782959244242904270, −1.48207310990774027324936125367, −0.873193878614773183523832811103,
0.873193878614773183523832811103, 1.48207310990774027324936125367, 2.61115291801782959244242904270, 3.50828238529535548574726346732, 3.94513573682298272413934743563, 4.13719132333362772741355965576, 4.74636824709949268335118783326, 5.17906672504472111908101207858, 6.35389253844449468960839880985, 6.46838322567371333364017657592, 7.00666108758363224630411901529, 7.45311496359253204808358138568, 7.70808153712614068374976575634, 8.375848115054907090016583855751, 8.763191837037744281387501238879, 9.129171517005568276859253511243, 9.766656387781431151823788433392, 10.03091677087469056227918960742, 10.71009106781184143203315927908, 10.91070189481907922956927318156
