| L(s) = 1 | − 20·31-s + 8·37-s − 20·41-s − 8·43-s + 10·49-s − 20·53-s + 24·67-s − 8·71-s − 28·79-s + 28·89-s + 8·107-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | − 3.59·31-s + 1.31·37-s − 3.12·41-s − 1.21·43-s + 10/7·49-s − 2.74·53-s + 2.93·67-s − 0.949·71-s − 3.15·79-s + 2.96·89-s + 0.773·107-s + 6/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4462112195\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4462112195\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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
−8.174724527848889956806089709828, −7.72245141839560348990795671873, −7.37275693547771654853789522853, −7.01524480659532236746056979592, −6.82572682119283573777787240605, −6.38453034787917308936478963289, −5.99489350071448969992445769950, −5.50049250889242634962875459496, −5.41136457983907222899504045179, −4.94693218148454056287421844696, −4.54387220811817044759358299572, −4.19761845725705193880022237990, −3.57631236054534981209029571192, −3.30975093088840062450932444575, −3.24108789999648633850882728161, −2.35321444672233488719118895608, −1.94932438853790802435305219186, −1.70213899085251737986528195714, −1.07037968262748505843695831373, −0.16654237439607242280517030264,
0.16654237439607242280517030264, 1.07037968262748505843695831373, 1.70213899085251737986528195714, 1.94932438853790802435305219186, 2.35321444672233488719118895608, 3.24108789999648633850882728161, 3.30975093088840062450932444575, 3.57631236054534981209029571192, 4.19761845725705193880022237990, 4.54387220811817044759358299572, 4.94693218148454056287421844696, 5.41136457983907222899504045179, 5.50049250889242634962875459496, 5.99489350071448969992445769950, 6.38453034787917308936478963289, 6.82572682119283573777787240605, 7.01524480659532236746056979592, 7.37275693547771654853789522853, 7.72245141839560348990795671873, 8.174724527848889956806089709828
