| L(s) = 1 | + 16·7-s + 28·17-s − 304·23-s − 138·25-s + 448·31-s − 140·41-s − 672·47-s − 494·49-s − 144·71-s + 588·73-s − 928·79-s + 532·89-s + 1.98e3·97-s − 2.35e3·103-s + 3.42e3·113-s + 448·119-s − 2.41e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s − 4.86e3·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.863·7-s + 0.399·17-s − 2.75·23-s − 1.10·25-s + 2.59·31-s − 0.533·41-s − 2.08·47-s − 1.44·49-s − 0.240·71-s + 0.942·73-s − 1.32·79-s + 0.633·89-s + 2.08·97-s − 2.24·103-s + 2.84·113-s + 0.345·119-s − 1.81·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s − 2.38·161-s + 0.000480·163-s + 0.000463·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 138 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 8 T + p^{3} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 2410 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 1594 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 14 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 12346 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 152 T + p^{3} T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 23578 T^{2} + p^{6} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 224 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 42058 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 70 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 33878 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 336 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 296746 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 125130 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 444890 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 571034 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 72 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 294 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 464 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 846522 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 266 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 994 T + p^{3} T^{2} )^{2} \) |
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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
−8.183924449172803788570407409082, −8.157858570179892080433455744861, −7.77636778673234223493897924216, −7.61868800074260553561676845739, −6.68930069004811913950486070934, −6.58528589900811511408875168801, −5.99593756437796060385705039761, −5.94863674875556486582052429206, −5.10711409356413635113510856554, −4.97393380522868250256558571517, −4.37568169389757031422688125671, −4.14485933025566698624011796493, −3.48489344499494139671822801320, −3.20572995311429181443952151534, −2.32764652496576824163262165300, −2.16409723567548868236602726302, −1.45386576889917842104640454744, −1.14180634486715889244267694988, 0, 0,
1.14180634486715889244267694988, 1.45386576889917842104640454744, 2.16409723567548868236602726302, 2.32764652496576824163262165300, 3.20572995311429181443952151534, 3.48489344499494139671822801320, 4.14485933025566698624011796493, 4.37568169389757031422688125671, 4.97393380522868250256558571517, 5.10711409356413635113510856554, 5.94863674875556486582052429206, 5.99593756437796060385705039761, 6.58528589900811511408875168801, 6.68930069004811913950486070934, 7.61868800074260553561676845739, 7.77636778673234223493897924216, 8.157858570179892080433455744861, 8.183924449172803788570407409082
