| L(s) = 1 | − 4·4-s − 4·5-s − 9·9-s + 32·11-s + 16·16-s + 38·19-s + 16·20-s − 109·25-s + 332·29-s + 288·31-s + 36·36-s − 384·41-s − 128·44-s + 36·45-s + 542·49-s − 128·55-s + 700·59-s − 300·61-s − 64·64-s + 264·71-s − 152·76-s + 2.65e3·79-s − 64·80-s + 81·81-s + 2.55e3·89-s − 152·95-s − 288·99-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.357·5-s − 1/3·9-s + 0.877·11-s + 1/4·16-s + 0.458·19-s + 0.178·20-s − 0.871·25-s + 2.12·29-s + 1.66·31-s + 1/6·36-s − 1.46·41-s − 0.438·44-s + 0.119·45-s + 1.58·49-s − 0.313·55-s + 1.54·59-s − 0.629·61-s − 1/8·64-s + 0.441·71-s − 0.229·76-s + 3.78·79-s − 0.0894·80-s + 1/9·81-s + 3.03·89-s − 0.164·95-s − 0.292·99-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324900 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.599405458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.599405458\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | $C_2$ | \( 1 + 4 T + p^{3} T^{2} \) |
| 19 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 542 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 16 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4138 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6910 T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 23758 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 166 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 144 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 92294 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 192 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 128738 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 150046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 291030 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 350 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 150 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 267098 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 132 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 482098 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 1328 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 827642 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1276 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1824190 T^{2} + p^{6} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.40319699045461139631830619880, −10.16036567350962865190002250454, −9.679758795970488439115959353858, −9.217744998971435977162630055950, −8.705303350136776325609326785670, −8.474492183974081899254945749605, −7.84748935151865632999444933852, −7.66584134299278639876583992507, −6.78742433100695187739116096321, −6.46213558782701628176957639091, −6.18460551684021598692542155600, −5.24437562588614932464437417698, −5.06039287209138093411735593468, −4.38554139500403145257117748309, −3.87674592623640586292066670312, −3.41201881770221327720848252105, −2.72221323473397649529301492712, −2.03396040363152498299613198435, −1.00057780030248702196824167022, −0.61474762406847611182360630605,
0.61474762406847611182360630605, 1.00057780030248702196824167022, 2.03396040363152498299613198435, 2.72221323473397649529301492712, 3.41201881770221327720848252105, 3.87674592623640586292066670312, 4.38554139500403145257117748309, 5.06039287209138093411735593468, 5.24437562588614932464437417698, 6.18460551684021598692542155600, 6.46213558782701628176957639091, 6.78742433100695187739116096321, 7.66584134299278639876583992507, 7.84748935151865632999444933852, 8.474492183974081899254945749605, 8.705303350136776325609326785670, 9.217744998971435977162630055950, 9.679758795970488439115959353858, 10.16036567350962865190002250454, 10.40319699045461139631830619880
