| L(s) = 1 | + 12.1·5-s + 22·7-s + 58.8·11-s − 49·13-s − 36.3·17-s + 70·19-s + 24.2·23-s + 22·25-s + 278.·29-s + 112·31-s + 266.·35-s + 281·37-s + 48.4·41-s − 50·43-s − 242.·47-s + 141·49-s − 374.·53-s + 714·55-s + 96.9·59-s − 679·61-s − 594.·65-s + 274·67-s − 446.·71-s − 511·73-s + 1.29e3·77-s + 526·79-s − 387.·83-s + ⋯ |
| L(s) = 1 | + 1.08·5-s + 1.18·7-s + 1.61·11-s − 1.04·13-s − 0.518·17-s + 0.845·19-s + 0.219·23-s + 0.175·25-s + 1.78·29-s + 0.648·31-s + 1.28·35-s + 1.24·37-s + 0.184·41-s − 0.177·43-s − 0.752·47-s + 0.411·49-s − 0.969·53-s + 1.75·55-s + 0.214·59-s − 1.42·61-s − 1.13·65-s + 0.499·67-s − 0.746·71-s − 0.819·73-s + 1.91·77-s + 0.749·79-s − 0.513·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.606525333\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.606525333\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 12.1T + 125T^{2} \) |
| 7 | \( 1 - 22T + 343T^{2} \) |
| 11 | \( 1 - 58.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 49T + 2.19e3T^{2} \) |
| 17 | \( 1 + 36.3T + 4.91e3T^{2} \) |
| 19 | \( 1 - 70T + 6.85e3T^{2} \) |
| 23 | \( 1 - 24.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 278.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 112T + 2.97e4T^{2} \) |
| 37 | \( 1 - 281T + 5.06e4T^{2} \) |
| 41 | \( 1 - 48.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 50T + 7.95e4T^{2} \) |
| 47 | \( 1 + 242.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 374.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 96.9T + 2.05e5T^{2} \) |
| 61 | \( 1 + 679T + 2.26e5T^{2} \) |
| 67 | \( 1 - 274T + 3.00e5T^{2} \) |
| 71 | \( 1 + 446.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 511T + 3.89e5T^{2} \) |
| 79 | \( 1 - 526T + 4.93e5T^{2} \) |
| 83 | \( 1 + 387.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 36.3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.77e3T + 9.12e5T^{2} \) |
| 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
−9.422252082508038207747673695129, −8.556739178549432726475761224190, −7.67546191465773307274830240483, −6.69601970059469025608153183706, −6.03582401044825461170377261949, −4.92471167268261940337148542039, −4.41377184386984531807823241693, −2.91308920979500858899167282335, −1.85124048382597322529656559640, −1.05641251931635380526575437433,
1.05641251931635380526575437433, 1.85124048382597322529656559640, 2.91308920979500858899167282335, 4.41377184386984531807823241693, 4.92471167268261940337148542039, 6.03582401044825461170377261949, 6.69601970059469025608153183706, 7.67546191465773307274830240483, 8.556739178549432726475761224190, 9.422252082508038207747673695129
