| L(s) = 1 | − 9-s + 4·17-s − 16·23-s + 6·25-s − 16·31-s + 12·41-s − 14·49-s + 16·71-s − 20·73-s + 16·79-s + 81-s + 12·89-s + 4·97-s + 32·103-s + 36·113-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 4·153-s + 157-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s + 0.970·17-s − 3.33·23-s + 6/5·25-s − 2.87·31-s + 1.87·41-s − 2·49-s + 1.89·71-s − 2.34·73-s + 1.80·79-s + 1/9·81-s + 1.27·89-s + 0.406·97-s + 3.15·103-s + 3.38·113-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 − 0.323·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 589824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.420877129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.420877129\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 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
−10.46588562974201334312723976408, −9.989173660185404835881376791360, −9.865580965114193933179141080449, −9.141097668822590516295160577972, −8.988519540765576842788830769250, −8.335445297652243570758118830512, −7.915609896852621107472486013912, −7.54453528470408688397465897336, −7.29608690080808381559727331837, −6.37883635359374734927061256576, −6.19498598648129175925372059556, −5.65893468317171426714674578773, −5.31668349965895694485825805748, −4.59377671535927490212946043161, −4.13624423366458036489930680012, −3.42010921896522317342307299368, −3.25553106959960841072741553326, −1.99038704802831102100480998231, −1.98975141550678631201287002043, −0.59281643592350509690463923604,
0.59281643592350509690463923604, 1.98975141550678631201287002043, 1.99038704802831102100480998231, 3.25553106959960841072741553326, 3.42010921896522317342307299368, 4.13624423366458036489930680012, 4.59377671535927490212946043161, 5.31668349965895694485825805748, 5.65893468317171426714674578773, 6.19498598648129175925372059556, 6.37883635359374734927061256576, 7.29608690080808381559727331837, 7.54453528470408688397465897336, 7.915609896852621107472486013912, 8.335445297652243570758118830512, 8.988519540765576842788830769250, 9.141097668822590516295160577972, 9.865580965114193933179141080449, 9.989173660185404835881376791360, 10.46588562974201334312723976408
