L(s) = 1 | + 1.56·2-s + 3-s + 0.458·4-s − 5-s + 1.56·6-s − 3.42·7-s − 2.41·8-s + 9-s − 1.56·10-s + 5.04·11-s + 0.458·12-s − 0.802·13-s − 5.36·14-s − 15-s − 4.70·16-s + 3.28·17-s + 1.56·18-s + 4.91·19-s − 0.458·20-s − 3.42·21-s + 7.91·22-s − 2.41·24-s + 25-s − 1.25·26-s + 27-s − 1.56·28-s − 5.37·29-s + ⋯ |
L(s) = 1 | + 1.10·2-s + 0.577·3-s + 0.229·4-s − 0.447·5-s + 0.640·6-s − 1.29·7-s − 0.854·8-s + 0.333·9-s − 0.495·10-s + 1.52·11-s + 0.132·12-s − 0.222·13-s − 1.43·14-s − 0.258·15-s − 1.17·16-s + 0.796·17-s + 0.369·18-s + 1.12·19-s − 0.102·20-s − 0.747·21-s + 1.68·22-s − 0.493·24-s + 0.200·25-s − 0.246·26-s + 0.192·27-s − 0.296·28-s − 0.998·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.199881954\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.199881954\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 7 | \( 1 + 3.42T + 7T^{2} \) |
| 11 | \( 1 - 5.04T + 11T^{2} \) |
| 13 | \( 1 + 0.802T + 13T^{2} \) |
| 17 | \( 1 - 3.28T + 17T^{2} \) |
| 19 | \( 1 - 4.91T + 19T^{2} \) |
| 29 | \( 1 + 5.37T + 29T^{2} \) |
| 31 | \( 1 + 2.57T + 31T^{2} \) |
| 37 | \( 1 + 8.00T + 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 + 0.373T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 - 3.23T + 53T^{2} \) |
| 59 | \( 1 - 8.14T + 59T^{2} \) |
| 61 | \( 1 - 7.79T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 2.40T + 71T^{2} \) |
| 73 | \( 1 - 3.31T + 73T^{2} \) |
| 79 | \( 1 - 9.28T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 + 0.377T + 89T^{2} \) |
| 97 | \( 1 - 12.9T + 97T^{2} \) |
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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
−7.67377688540789450520566461051, −6.85246186046503710921618186420, −6.58603306023657409899404962758, −5.56462370528438241505180333971, −5.07796752746785161206229342855, −3.87327529976654721379967221307, −3.63258111893400880809329286950, −3.21768782126855804445762296674, −2.07292294414274574313468770839, −0.72010337191029619575912190255,
0.72010337191029619575912190255, 2.07292294414274574313468770839, 3.21768782126855804445762296674, 3.63258111893400880809329286950, 3.87327529976654721379967221307, 5.07796752746785161206229342855, 5.56462370528438241505180333971, 6.58603306023657409899404962758, 6.85246186046503710921618186420, 7.67377688540789450520566461051