L(s) = 1 | + 3·3-s + 5·5-s + 20.2·7-s + 9·9-s − 7.79·11-s − 0.515·13-s + 15·15-s − 117.·17-s − 122.·19-s + 60.8·21-s − 23·23-s + 25·25-s + 27·27-s − 152.·29-s − 300.·31-s − 23.3·33-s + 101.·35-s − 241.·37-s − 1.54·39-s − 146.·41-s − 250.·43-s + 45·45-s + 212.·47-s + 68.5·49-s − 351.·51-s − 42.4·53-s − 38.9·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.09·7-s + 0.333·9-s − 0.213·11-s − 0.0109·13-s + 0.258·15-s − 1.67·17-s − 1.47·19-s + 0.632·21-s − 0.208·23-s + 0.200·25-s + 0.192·27-s − 0.973·29-s − 1.74·31-s − 0.123·33-s + 0.489·35-s − 1.07·37-s − 0.00634·39-s − 0.558·41-s − 0.887·43-s + 0.149·45-s + 0.659·47-s + 0.199·49-s − 0.964·51-s − 0.110·53-s − 0.0954·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 + 23T \) |
good | 7 | \( 1 - 20.2T + 343T^{2} \) |
| 11 | \( 1 + 7.79T + 1.33e3T^{2} \) |
| 13 | \( 1 + 0.515T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 122.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 152.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 300.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 241.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 146.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 250.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 212.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 42.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 166.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 229.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 362.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 624.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 519.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 917.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.14e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.21e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.05e3T + 9.12e5T^{2} \) |
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
−8.778265522359010818506847701973, −8.171818607007359044556874865214, −7.22681309299912312273188880173, −6.44273694878698523353029240794, −5.33010394386411212433987227087, −4.54599122802424149548645957867, −3.64306409216466391536032476142, −2.15715136633265872776571685506, −1.82834356729317036315420898610, 0,
1.82834356729317036315420898610, 2.15715136633265872776571685506, 3.64306409216466391536032476142, 4.54599122802424149548645957867, 5.33010394386411212433987227087, 6.44273694878698523353029240794, 7.22681309299912312273188880173, 8.171818607007359044556874865214, 8.778265522359010818506847701973