L(s) = 1 | − 2·3-s + 2·5-s − 2·7-s + 3·9-s + 2·13-s − 4·15-s − 6·17-s + 4·21-s + 3·25-s − 4·27-s − 16·29-s + 4·31-s − 4·35-s + 8·37-s − 4·39-s − 16·41-s + 14·43-s + 6·45-s + 4·47-s + 2·49-s + 12·51-s + 4·53-s + 16·59-s − 6·63-s + 4·65-s − 8·67-s + 6·73-s + ⋯ |
L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.755·7-s + 9-s + 0.554·13-s − 1.03·15-s − 1.45·17-s + 0.872·21-s + 3/5·25-s − 0.769·27-s − 2.97·29-s + 0.718·31-s − 0.676·35-s + 1.31·37-s − 0.640·39-s − 2.49·41-s + 2.13·43-s + 0.894·45-s + 0.583·47-s + 2/7·49-s + 1.68·51-s + 0.549·53-s + 2.08·59-s − 0.755·63-s + 0.496·65-s − 0.977·67-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 52707600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 52707600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.024297523\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.024297523\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
good | 7 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 2 T + 14 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 8 T + 38 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 14 T + 122 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 66 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 6 T + 142 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 8 T + 122 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 14 T + 202 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−7.82320835104598924970682743895, −7.79772267354123084841183882330, −7.26829962682882809701059513233, −6.83560281143891450324508436511, −6.58472583114116212325567953339, −6.44621004574518776955165635159, −5.90836766197025383736904665397, −5.73399823425084965229398351847, −5.31239497579882299638247191092, −5.18074565250516285379607546176, −4.42904352298157242178784842768, −4.28847331042126335626868953854, −3.78755018541482723469521419395, −3.49412432523900960905248524080, −2.86832342819627180621586755826, −2.35926172764118332566370823746, −1.92205604086701977235384780140, −1.69404031204909605069835299932, −0.67805290625755417885996539903, −0.56176964179445401708832642983,
0.56176964179445401708832642983, 0.67805290625755417885996539903, 1.69404031204909605069835299932, 1.92205604086701977235384780140, 2.35926172764118332566370823746, 2.86832342819627180621586755826, 3.49412432523900960905248524080, 3.78755018541482723469521419395, 4.28847331042126335626868953854, 4.42904352298157242178784842768, 5.18074565250516285379607546176, 5.31239497579882299638247191092, 5.73399823425084965229398351847, 5.90836766197025383736904665397, 6.44621004574518776955165635159, 6.58472583114116212325567953339, 6.83560281143891450324508436511, 7.26829962682882809701059513233, 7.79772267354123084841183882330, 7.82320835104598924970682743895