| L(s) = 1 | − 3-s − 2·5-s − 4·13-s + 2·15-s − 6·17-s + 4·19-s + 4·23-s + 5·25-s + 27-s + 12·29-s + 8·31-s + 10·37-s + 4·39-s − 20·41-s + 24·43-s + 8·47-s + 6·51-s − 6·53-s − 4·57-s − 4·59-s + 10·61-s + 8·65-s − 12·67-s − 4·69-s + 8·71-s − 2·73-s − 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s − 1.10·13-s + 0.516·15-s − 1.45·17-s + 0.917·19-s + 0.834·23-s + 25-s + 0.192·27-s + 2.22·29-s + 1.43·31-s + 1.64·37-s + 0.640·39-s − 3.12·41-s + 3.65·43-s + 1.16·47-s + 0.840·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s + 1.28·61-s + 0.992·65-s − 1.46·67-s − 0.481·69-s + 0.949·71-s − 0.234·73-s − 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.203521410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.203521410\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T - T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6 T - 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
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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
−9.975742013395195839481519038453, −9.602816177979804266907907184148, −9.101000062804199643697858866918, −8.787613724494123880557219642648, −8.338294023965932215726266706558, −7.79019533780240422032518805608, −7.64379771519821630207919822608, −6.90192292088281201617879618408, −6.72001794879956121551621364488, −6.43501688408396885553384576504, −5.65448227013864223952672249250, −5.29298328694348194444707909422, −4.58013903155345666208524405091, −4.56377360348747265486324767540, −4.11146610040708922725931001698, −3.14311910170044464725502615136, −2.77981929006356827216035258943, −2.38885058078606812271569220319, −1.15553536961944291464491267818, −0.57430000642020008742110684254,
0.57430000642020008742110684254, 1.15553536961944291464491267818, 2.38885058078606812271569220319, 2.77981929006356827216035258943, 3.14311910170044464725502615136, 4.11146610040708922725931001698, 4.56377360348747265486324767540, 4.58013903155345666208524405091, 5.29298328694348194444707909422, 5.65448227013864223952672249250, 6.43501688408396885553384576504, 6.72001794879956121551621364488, 6.90192292088281201617879618408, 7.64379771519821630207919822608, 7.79019533780240422032518805608, 8.338294023965932215726266706558, 8.787613724494123880557219642648, 9.101000062804199643697858866918, 9.602816177979804266907907184148, 9.975742013395195839481519038453
