| L(s) = 1 | − 3-s + 0.169·5-s − 1.46·7-s + 9-s − 11-s + 13-s − 0.169·15-s + 6.72·17-s − 6.80·19-s + 1.46·21-s + 5.29·23-s − 4.97·25-s − 27-s − 4.23·29-s − 7.17·31-s + 33-s − 0.247·35-s + 5.03·37-s − 39-s + 0.720·41-s + 12.1·43-s + 0.169·45-s − 10.4·47-s − 4.86·49-s − 6.72·51-s + 12.9·53-s − 0.169·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.0757·5-s − 0.552·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s − 0.0437·15-s + 1.63·17-s − 1.56·19-s + 0.319·21-s + 1.10·23-s − 0.994·25-s − 0.192·27-s − 0.785·29-s − 1.28·31-s + 0.174·33-s − 0.0418·35-s + 0.826·37-s − 0.160·39-s + 0.112·41-s + 1.85·43-s + 0.0252·45-s − 1.52·47-s − 0.694·49-s − 0.941·51-s + 1.77·53-s − 0.0228·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 0.169T + 5T^{2} \) |
| 7 | \( 1 + 1.46T + 7T^{2} \) |
| 17 | \( 1 - 6.72T + 17T^{2} \) |
| 19 | \( 1 + 6.80T + 19T^{2} \) |
| 23 | \( 1 - 5.29T + 23T^{2} \) |
| 29 | \( 1 + 4.23T + 29T^{2} \) |
| 31 | \( 1 + 7.17T + 31T^{2} \) |
| 37 | \( 1 - 5.03T + 37T^{2} \) |
| 41 | \( 1 - 0.720T + 41T^{2} \) |
| 43 | \( 1 - 12.1T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 3.92T + 59T^{2} \) |
| 61 | \( 1 - 1.23T + 61T^{2} \) |
| 67 | \( 1 - 0.504T + 67T^{2} \) |
| 71 | \( 1 + 12.0T + 71T^{2} \) |
| 73 | \( 1 + 3.12T + 73T^{2} \) |
| 79 | \( 1 - 6.05T + 79T^{2} \) |
| 83 | \( 1 + 16.0T + 83T^{2} \) |
| 89 | \( 1 + 1.80T + 89T^{2} \) |
| 97 | \( 1 + 9.83T + 97T^{2} \) |
| 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
−8.137835949207435681876008937630, −7.43231342925013927912375407159, −6.70098026346860369000911448387, −5.79541785783688681667334048231, −5.48310293043041551029111179595, −4.30074972390719400148492420310, −3.58978401908644876378169396075, −2.54143346477786049840515085453, −1.33968086034716924630983028067, 0,
1.33968086034716924630983028067, 2.54143346477786049840515085453, 3.58978401908644876378169396075, 4.30074972390719400148492420310, 5.48310293043041551029111179595, 5.79541785783688681667334048231, 6.70098026346860369000911448387, 7.43231342925013927912375407159, 8.137835949207435681876008937630
