| L(s) = 1 | − 2·4-s + 22·7-s − 14·13-s + 4·16-s + 58·19-s − 44·28-s + 58·31-s + 112·37-s + 10·43-s + 265·49-s + 28·52-s − 110·61-s − 8·64-s − 74·67-s − 32·73-s − 116·76-s + 208·79-s − 308·91-s + 82·97-s + 124·103-s − 338·109-s + 88·112-s + 224·121-s − 116·124-s + 127-s + 131-s + 1.27e3·133-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 22/7·7-s − 1.07·13-s + 1/4·16-s + 3.05·19-s − 1.57·28-s + 1.87·31-s + 3.02·37-s + 0.232·43-s + 5.40·49-s + 7/13·52-s − 1.80·61-s − 1/8·64-s − 1.10·67-s − 0.438·73-s − 1.52·76-s + 2.63·79-s − 3.38·91-s + 0.845·97-s + 1.20·103-s − 3.10·109-s + 0.785·112-s + 1.85·121-s − 0.935·124-s + 0.00787·127-s + 0.00763·131-s + 9.59·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.844771175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.844771175\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 11 T + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 224 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 416 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 400 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 496 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 29 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 56 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )( 1 + 46 T + p^{2} T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 368 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 1010 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6080 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 55 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 37 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8930 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 16 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 104 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 12896 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 2590 T^{2} + p^{4} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 41 T + p^{2} T^{2} )^{2} \) |
| 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
−11.07587468539634047882876096632, −10.91190681899679311907956805795, −9.969099601345732355573114851548, −9.953446801344310271357201146630, −9.171681433700481810244698548795, −8.963481527929195873201781000030, −8.058803801815386675599556849632, −7.956650135646942926015841928265, −7.50937152179266410801663315369, −7.46458390543129296968117186282, −6.31740727865913897819686444179, −5.75035823508242617876121062642, −5.05703326142690238016295005176, −4.88083662238282463498675877327, −4.62409585325749933966654471139, −3.89642476574519607506994787436, −2.86920028031834839988941187287, −2.36289908522304622447091022962, −1.22560877048394692456951885028, −1.08301972131667287789581246218,
1.08301972131667287789581246218, 1.22560877048394692456951885028, 2.36289908522304622447091022962, 2.86920028031834839988941187287, 3.89642476574519607506994787436, 4.62409585325749933966654471139, 4.88083662238282463498675877327, 5.05703326142690238016295005176, 5.75035823508242617876121062642, 6.31740727865913897819686444179, 7.46458390543129296968117186282, 7.50937152179266410801663315369, 7.956650135646942926015841928265, 8.058803801815386675599556849632, 8.963481527929195873201781000030, 9.171681433700481810244698548795, 9.953446801344310271357201146630, 9.969099601345732355573114851548, 10.91190681899679311907956805795, 11.07587468539634047882876096632
