L(s) = 1 | + 2·2-s + 4·4-s + 16.6·5-s − 7.94·7-s + 8·8-s + 33.2·10-s − 30.4·13-s − 15.8·14-s + 16·16-s − 122.·17-s − 152.·19-s + 66.4·20-s + 132.·23-s + 150.·25-s − 60.9·26-s − 31.7·28-s + 93.3·29-s + 88.3·31-s + 32·32-s − 244.·34-s − 132·35-s − 219.·37-s − 305.·38-s + 132.·40-s − 95.8·41-s − 371.·43-s + 265.·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.48·5-s − 0.429·7-s + 0.353·8-s + 1.05·10-s − 0.649·13-s − 0.303·14-s + 0.250·16-s − 1.74·17-s − 1.84·19-s + 0.742·20-s + 1.20·23-s + 1.20·25-s − 0.459·26-s − 0.214·28-s + 0.597·29-s + 0.511·31-s + 0.176·32-s − 1.23·34-s − 0.637·35-s − 0.975·37-s − 1.30·38-s + 0.525·40-s − 0.365·41-s − 1.31·43-s + 0.851·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2178 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 5 | \( 1 - 16.6T + 125T^{2} \) |
| 7 | \( 1 + 7.94T + 343T^{2} \) |
| 13 | \( 1 + 30.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 122.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 152.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 132.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 93.3T + 2.43e4T^{2} \) |
| 31 | \( 1 - 88.3T + 2.97e4T^{2} \) |
| 37 | \( 1 + 219.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 95.8T + 6.89e4T^{2} \) |
| 43 | \( 1 + 371.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 95.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 62.6T + 1.48e5T^{2} \) |
| 59 | \( 1 + 99.8T + 2.05e5T^{2} \) |
| 61 | \( 1 + 615.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 1.04e3T + 3.00e5T^{2} \) |
| 71 | \( 1 - 94.9T + 3.57e5T^{2} \) |
| 73 | \( 1 + 459.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 646.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 434.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.62e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 48.9T + 9.12e5T^{2} \) |
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
−8.567251881014475898415784934536, −7.17702953227153154119847340842, −6.40285492592813223557313470032, −6.21870026241637005072805456052, −4.93467038183076916218901526467, −4.59483623579764863604239214759, −3.19621815622283191073250692079, −2.34051421425134054339883025659, −1.69576599012176674458852867614, 0,
1.69576599012176674458852867614, 2.34051421425134054339883025659, 3.19621815622283191073250692079, 4.59483623579764863604239214759, 4.93467038183076916218901526467, 6.21870026241637005072805456052, 6.40285492592813223557313470032, 7.17702953227153154119847340842, 8.567251881014475898415784934536