L(s) = 1 | − 0.765·2-s − 1.41·4-s − 0.0823·5-s − 2.76·7-s + 2.61·8-s + 0.0630·10-s + 0.668·11-s − 3.28·13-s + 2.11·14-s + 0.828·16-s + 3.64·19-s + 0.116·20-s − 0.511·22-s + 9.30·23-s − 4.99·25-s + 2.51·26-s + 3.91·28-s + 6.24·29-s − 5.04·31-s − 5.86·32-s + 0.227·35-s − 2.40·37-s − 2.78·38-s − 0.215·40-s − 0.480·41-s + 8.27·43-s − 0.944·44-s + ⋯ |
L(s) = 1 | − 0.541·2-s − 0.707·4-s − 0.0368·5-s − 1.04·7-s + 0.923·8-s + 0.0199·10-s + 0.201·11-s − 0.910·13-s + 0.565·14-s + 0.207·16-s + 0.835·19-s + 0.0260·20-s − 0.109·22-s + 1.94·23-s − 0.998·25-s + 0.492·26-s + 0.739·28-s + 1.15·29-s − 0.905·31-s − 1.03·32-s + 0.0385·35-s − 0.395·37-s − 0.451·38-s − 0.0340·40-s − 0.0749·41-s + 1.26·43-s − 0.142·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 + 0.765T + 2T^{2} \) |
| 5 | \( 1 + 0.0823T + 5T^{2} \) |
| 7 | \( 1 + 2.76T + 7T^{2} \) |
| 11 | \( 1 - 0.668T + 11T^{2} \) |
| 13 | \( 1 + 3.28T + 13T^{2} \) |
| 19 | \( 1 - 3.64T + 19T^{2} \) |
| 23 | \( 1 - 9.30T + 23T^{2} \) |
| 29 | \( 1 - 6.24T + 29T^{2} \) |
| 31 | \( 1 + 5.04T + 31T^{2} \) |
| 37 | \( 1 + 2.40T + 37T^{2} \) |
| 41 | \( 1 + 0.480T + 41T^{2} \) |
| 43 | \( 1 - 8.27T + 43T^{2} \) |
| 47 | \( 1 - 8.88T + 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 9.59T + 59T^{2} \) |
| 61 | \( 1 - 2.58T + 61T^{2} \) |
| 67 | \( 1 + 0.944T + 67T^{2} \) |
| 71 | \( 1 + 3.28T + 71T^{2} \) |
| 73 | \( 1 + 15.6T + 73T^{2} \) |
| 79 | \( 1 - 8.37T + 79T^{2} \) |
| 83 | \( 1 + 0.899T + 83T^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.783926517448200148183450702365, −7.61422533789022732825787409520, −7.26097475694320396442931580358, −6.25067105840251398932134681411, −5.29522271460068380067330445346, −4.57759223355770108440913130420, −3.56931942484959236934649365329, −2.73103065078823272921756603439, −1.21205278916614697690086478316, 0,
1.21205278916614697690086478316, 2.73103065078823272921756603439, 3.56931942484959236934649365329, 4.57759223355770108440913130420, 5.29522271460068380067330445346, 6.25067105840251398932134681411, 7.26097475694320396442931580358, 7.61422533789022732825787409520, 8.783926517448200148183450702365