| L(s) = 1 | − 9-s + 12·17-s + 16·23-s − 25-s − 8·31-s − 4·41-s − 16·47-s − 14·49-s + 16·71-s + 28·73-s + 24·79-s + 81-s + 28·89-s + 4·97-s + 16·103-s − 20·113-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 12·153-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 2.91·17-s + 3.33·23-s − 1/5·25-s − 1.43·31-s − 0.624·41-s − 2.33·47-s − 2·49-s + 1.89·71-s + 3.27·73-s + 2.70·79-s + 1/9·81-s + 2.96·89-s + 0.406·97-s + 1.57·103-s − 1.88·113-s + 2·121-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 − 0.970·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14745600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.651951018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.651951018\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.669941201542239558134397682084, −8.130291947411410281772056904157, −7.897620318788888414399455137568, −7.76196715124696625764126378178, −7.21676501343697961242788980754, −6.80793980648533202355979558551, −6.44905454539681158935985290709, −6.24561977399359378496670567404, −5.41279020010056978274654825318, −5.29591713244453053295412388127, −4.93857315295163951926553406596, −4.89302064282077834299802456525, −3.86888290256714392128786250981, −3.34628195851288466550937329442, −3.28794000473019828584181593356, −3.11109220898003203078983713960, −2.09303570358724683866335201499, −1.78581267250210453233796024109, −0.941266393337892394198209854188, −0.72136183299826268589516836633,
0.72136183299826268589516836633, 0.941266393337892394198209854188, 1.78581267250210453233796024109, 2.09303570358724683866335201499, 3.11109220898003203078983713960, 3.28794000473019828584181593356, 3.34628195851288466550937329442, 3.86888290256714392128786250981, 4.89302064282077834299802456525, 4.93857315295163951926553406596, 5.29591713244453053295412388127, 5.41279020010056978274654825318, 6.24561977399359378496670567404, 6.44905454539681158935985290709, 6.80793980648533202355979558551, 7.21676501343697961242788980754, 7.76196715124696625764126378178, 7.897620318788888414399455137568, 8.130291947411410281772056904157, 8.669941201542239558134397682084
