| L(s) = 1 | + 2·2-s + 2·3-s + 4-s + 4·6-s + 3·9-s + 2·12-s + 2·13-s + 16-s − 4·17-s + 6·18-s − 8·23-s − 2·25-s + 4·26-s + 4·27-s − 4·29-s − 8·31-s − 2·32-s − 8·34-s + 3·36-s − 4·37-s + 4·39-s − 16·41-s − 8·43-s − 16·46-s − 12·47-s + 2·48-s − 6·49-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 1/2·4-s + 1.63·6-s + 9-s + 0.577·12-s + 0.554·13-s + 1/4·16-s − 0.970·17-s + 1.41·18-s − 1.66·23-s − 2/5·25-s + 0.784·26-s + 0.769·27-s − 0.742·29-s − 1.43·31-s − 0.353·32-s − 1.37·34-s + 1/2·36-s − 0.657·37-s + 0.640·39-s − 2.49·41-s − 1.21·43-s − 2.35·46-s − 1.75·47-s + 0.288·48-s − 6/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22268961 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22268961 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.724566047\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.724566047\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| 13 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 138 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 12 T + 98 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 142 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 150 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T + 138 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 166 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.325843101801022614943319996593, −8.184600015723720421768744736661, −7.79671947954750849685991601850, −7.38794124670117779089621623724, −6.98954663583003255894451219393, −6.55143768682375020576517924147, −6.34586045957054567496433190938, −5.86813260563745771979075849094, −5.30312332818103214090616537023, −5.13974455914584362946998514267, −4.66622302432326515315077961036, −4.36856280910379352197870984596, −3.79145881279096002479088164328, −3.68995078116544685842293681117, −3.23124387164511824411387352811, −3.07176862226463899349375807098, −1.91781357271169463443113911978, −1.87595027443132049659746198560, −1.76642041014573511776794183874, −0.34015431765544724061073791039,
0.34015431765544724061073791039, 1.76642041014573511776794183874, 1.87595027443132049659746198560, 1.91781357271169463443113911978, 3.07176862226463899349375807098, 3.23124387164511824411387352811, 3.68995078116544685842293681117, 3.79145881279096002479088164328, 4.36856280910379352197870984596, 4.66622302432326515315077961036, 5.13974455914584362946998514267, 5.30312332818103214090616537023, 5.86813260563745771979075849094, 6.34586045957054567496433190938, 6.55143768682375020576517924147, 6.98954663583003255894451219393, 7.38794124670117779089621623724, 7.79671947954750849685991601850, 8.184600015723720421768744736661, 8.325843101801022614943319996593
