| L(s) = 1 | − 3.12·3-s + 21.5·5-s + 11.5·7-s − 17.2·9-s − 34.5·11-s + 13·13-s − 67.2·15-s + 41.6·17-s − 26.0·19-s − 36.1·21-s − 96.2·23-s + 339.·25-s + 138.·27-s − 218.·29-s − 144.·31-s + 107.·33-s + 249.·35-s − 122.·37-s − 40.5·39-s − 163.·41-s − 407.·43-s − 372.·45-s − 107.·47-s − 209.·49-s − 129.·51-s + 103.·53-s − 745.·55-s + ⋯ |
| L(s) = 1 | − 0.600·3-s + 1.92·5-s + 0.624·7-s − 0.639·9-s − 0.947·11-s + 0.277·13-s − 1.15·15-s + 0.594·17-s − 0.314·19-s − 0.375·21-s − 0.872·23-s + 2.71·25-s + 0.984·27-s − 1.40·29-s − 0.836·31-s + 0.569·33-s + 1.20·35-s − 0.544·37-s − 0.166·39-s − 0.621·41-s − 1.44·43-s − 1.23·45-s − 0.332·47-s − 0.609·49-s − 0.356·51-s + 0.268·53-s − 1.82·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1664 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - 13T \) |
| good | 3 | \( 1 + 3.12T + 27T^{2} \) |
| 5 | \( 1 - 21.5T + 125T^{2} \) |
| 7 | \( 1 - 11.5T + 343T^{2} \) |
| 11 | \( 1 + 34.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 41.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.0T + 6.85e3T^{2} \) |
| 23 | \( 1 + 96.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 218.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 144.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 122.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 163.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 407.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 107.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 103.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 652.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 293.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 241.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 117.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 284.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.11e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 820.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 356.T + 9.12e5T^{2} \) |
| 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
−8.679315299929419694019781685755, −7.892968224109610173670752873870, −6.74239900985189310838183860998, −5.93904281302604214814765456790, −5.43577275742719312656410413991, −4.91630456893731092326248073361, −3.30355714930646693575361685300, −2.19582407456884385575641664870, −1.51670664409161477104939715607, 0,
1.51670664409161477104939715607, 2.19582407456884385575641664870, 3.30355714930646693575361685300, 4.91630456893731092326248073361, 5.43577275742719312656410413991, 5.93904281302604214814765456790, 6.74239900985189310838183860998, 7.892968224109610173670752873870, 8.679315299929419694019781685755
