| L(s) = 1 | + 2·2-s + 3·4-s − 5-s + 2·7-s + 4·8-s − 2·10-s − 7·11-s + 6·13-s + 4·14-s + 5·16-s + 4·17-s − 2·19-s − 3·20-s − 14·22-s − 2·23-s + 2·25-s + 12·26-s + 6·28-s + 11·29-s + 8·31-s + 6·32-s + 8·34-s − 2·35-s + 3·37-s − 4·38-s − 4·40-s + 5·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.447·5-s + 0.755·7-s + 1.41·8-s − 0.632·10-s − 2.11·11-s + 1.66·13-s + 1.06·14-s + 5/4·16-s + 0.970·17-s − 0.458·19-s − 0.670·20-s − 2.98·22-s − 0.417·23-s + 2/5·25-s + 2.35·26-s + 1.13·28-s + 2.04·29-s + 1.43·31-s + 1.06·32-s + 1.37·34-s − 0.338·35-s + 0.493·37-s − 0.648·38-s − 0.632·40-s + 0.780·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.693905730\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.693905730\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| 19 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 7 T + 3 p T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 6 T + 30 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 11 T + 77 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 8 T + 58 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 3 T + 65 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 5 T + 27 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 13 T + 127 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - T + 83 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 25 T + 261 T^{2} - 25 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 11 T + 137 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 11 T + 91 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 58 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 3 T - 67 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 10 T + 166 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 127 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 5 T + 183 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 183 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−8.735912427785060027200071700491, −8.572417015715651033150574995106, −8.394723335542688423890482660872, −7.943267884736368674291763708879, −7.66560285481341894954541045845, −7.19989190130707047104776269913, −6.66757177527337767734633191359, −6.45455863470615685828068501095, −5.71953682293020473232997294757, −5.69059916047118335369982586535, −5.14738887025707870184973032707, −4.86345975302589407860331433156, −4.29666128466038975278772245658, −4.06289763526495944216964580084, −3.47280229940645882563311363422, −3.04421093520902745252441482350, −2.45003777023784668631357170395, −2.32795393932379781245519563327, −1.24999027785414522844343316450, −0.822128709153847757450929443586,
0.822128709153847757450929443586, 1.24999027785414522844343316450, 2.32795393932379781245519563327, 2.45003777023784668631357170395, 3.04421093520902745252441482350, 3.47280229940645882563311363422, 4.06289763526495944216964580084, 4.29666128466038975278772245658, 4.86345975302589407860331433156, 5.14738887025707870184973032707, 5.69059916047118335369982586535, 5.71953682293020473232997294757, 6.45455863470615685828068501095, 6.66757177527337767734633191359, 7.19989190130707047104776269913, 7.66560285481341894954541045845, 7.943267884736368674291763708879, 8.394723335542688423890482660872, 8.572417015715651033150574995106, 8.735912427785060027200071700491
