| L(s) = 1 | + 3-s − 5-s + 7-s − 6·11-s − 6·13-s − 15-s + 4·17-s + 19-s + 21-s − 4·23-s − 27-s − 16·29-s + 31-s − 6·33-s − 35-s − 7·37-s − 6·39-s − 12·41-s − 2·43-s + 2·47-s − 6·49-s + 4·51-s − 4·53-s + 6·55-s + 57-s − 8·59-s + 14·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s − 1.80·11-s − 1.66·13-s − 0.258·15-s + 0.970·17-s + 0.229·19-s + 0.218·21-s − 0.834·23-s − 0.192·27-s − 2.97·29-s + 0.179·31-s − 1.04·33-s − 0.169·35-s − 1.15·37-s − 0.960·39-s − 1.87·41-s − 0.304·43-s + 0.291·47-s − 6/7·49-s + 0.560·51-s − 0.549·53-s + 0.809·55-s + 0.132·57-s − 1.04·59-s + 1.79·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2822400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2242359689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2242359689\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 4 T - T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - T - 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 7 T - 18 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T - 72 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 12 T + 55 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−9.799017111026352812775492126586, −8.916382870521686932449070626475, −8.911610802079641945430882371073, −8.200444409059272060884939297142, −7.77083021382739542142480606847, −7.71044809374244174557197014840, −7.48584219627726846980685576209, −6.89937541278543341700318299896, −6.45253632322690310653945760658, −5.53813185051108553750958509458, −5.50308691489441016538276865499, −5.08384267299640161839052252141, −4.74646859968446338972797107605, −3.94402396756791447706598707449, −3.63127476110818940837555110046, −3.06330121276965694775061512739, −2.65133548171442626748545072215, −1.98534803725846908512535637551, −1.67127185103252551529350466433, −0.16023828338465719120110060703,
0.16023828338465719120110060703, 1.67127185103252551529350466433, 1.98534803725846908512535637551, 2.65133548171442626748545072215, 3.06330121276965694775061512739, 3.63127476110818940837555110046, 3.94402396756791447706598707449, 4.74646859968446338972797107605, 5.08384267299640161839052252141, 5.50308691489441016538276865499, 5.53813185051108553750958509458, 6.45253632322690310653945760658, 6.89937541278543341700318299896, 7.48584219627726846980685576209, 7.71044809374244174557197014840, 7.77083021382739542142480606847, 8.200444409059272060884939297142, 8.911610802079641945430882371073, 8.916382870521686932449070626475, 9.799017111026352812775492126586
