| L(s) = 1 | − 2·3-s + 5·7-s + 9-s + 3·11-s − 13-s − 3·17-s + 4·19-s − 10·21-s + 6·23-s + 4·27-s + 9·29-s − 5·31-s − 6·33-s − 2·37-s + 2·39-s + 2·43-s − 9·47-s + 18·49-s + 6·51-s + 9·53-s − 8·57-s + 9·59-s − 61-s + 5·63-s + 5·67-s − 12·69-s − 14·73-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1.88·7-s + 1/3·9-s + 0.904·11-s − 0.277·13-s − 0.727·17-s + 0.917·19-s − 2.18·21-s + 1.25·23-s + 0.769·27-s + 1.67·29-s − 0.898·31-s − 1.04·33-s − 0.328·37-s + 0.320·39-s + 0.304·43-s − 1.31·47-s + 18/7·49-s + 0.840·51-s + 1.23·53-s − 1.05·57-s + 1.17·59-s − 0.128·61-s + 0.629·63-s + 0.610·67-s − 1.44·69-s − 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.807804361\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.807804361\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 9 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.365065719231970696713646552259, −7.23947614407772839833597241376, −6.88117473141994835586432532948, −5.91092333464382779108395360905, −5.14172657030758601942176541313, −4.83085076271036741451947245920, −4.02193344290000431177600613182, −2.73552908053811080293570367600, −1.57711598907063564206945008200, −0.852657327615444012964453501572,
0.852657327615444012964453501572, 1.57711598907063564206945008200, 2.73552908053811080293570367600, 4.02193344290000431177600613182, 4.83085076271036741451947245920, 5.14172657030758601942176541313, 5.91092333464382779108395360905, 6.88117473141994835586432532948, 7.23947614407772839833597241376, 8.365065719231970696713646552259
