L(s) = 1 | + 0.0849·2-s + 1.40·3-s − 1.99·4-s − 3.29·5-s + 0.118·6-s − 0.339·8-s − 1.03·9-s − 0.279·10-s + 1.30·11-s − 2.78·12-s − 2.58·13-s − 4.61·15-s + 3.95·16-s − 5.10·17-s − 0.0883·18-s − 6.41·19-s + 6.56·20-s + 0.111·22-s − 5.61·23-s − 0.474·24-s + 5.85·25-s − 0.219·26-s − 5.65·27-s + 5.85·29-s − 0.391·30-s + 5.23·31-s + 1.01·32-s + ⋯ |
L(s) = 1 | + 0.0600·2-s + 0.808·3-s − 0.996·4-s − 1.47·5-s + 0.0485·6-s − 0.119·8-s − 0.346·9-s − 0.0884·10-s + 0.394·11-s − 0.805·12-s − 0.717·13-s − 1.19·15-s + 0.989·16-s − 1.23·17-s − 0.0208·18-s − 1.47·19-s + 1.46·20-s + 0.0237·22-s − 1.17·23-s − 0.0969·24-s + 1.17·25-s − 0.0431·26-s − 1.08·27-s + 1.08·29-s − 0.0715·30-s + 0.939·31-s + 0.179·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
good | 2 | \( 1 - 0.0849T + 2T^{2} \) |
| 3 | \( 1 - 1.40T + 3T^{2} \) |
| 5 | \( 1 + 3.29T + 5T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 + 2.58T + 13T^{2} \) |
| 17 | \( 1 + 5.10T + 17T^{2} \) |
| 19 | \( 1 + 6.41T + 19T^{2} \) |
| 23 | \( 1 + 5.61T + 23T^{2} \) |
| 29 | \( 1 - 5.85T + 29T^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 - 8.05T + 37T^{2} \) |
| 41 | \( 1 - 1.40T + 41T^{2} \) |
| 43 | \( 1 - 3.50T + 43T^{2} \) |
| 47 | \( 1 - 4.81T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 4.83T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 8.93T + 67T^{2} \) |
| 71 | \( 1 + 7.60T + 71T^{2} \) |
| 73 | \( 1 + 3.59T + 73T^{2} \) |
| 79 | \( 1 + 7.58T + 79T^{2} \) |
| 83 | \( 1 + 5.15T + 83T^{2} \) |
| 89 | \( 1 + 3.39T + 89T^{2} \) |
| 97 | \( 1 + 9.10T + 97T^{2} \) |
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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
−11.17419280829796344723593665420, −9.970172699839713454897958868496, −8.868759949190761444541453318371, −8.370043843735109856621665771160, −7.63414446748391312982890904811, −6.24131039262292148742319840269, −4.47531390086769729583321947737, −4.09346514729477623820244408824, −2.70820989155748173056969903571, 0,
2.70820989155748173056969903571, 4.09346514729477623820244408824, 4.47531390086769729583321947737, 6.24131039262292148742319840269, 7.63414446748391312982890904811, 8.370043843735109856621665771160, 8.868759949190761444541453318371, 9.970172699839713454897958868496, 11.17419280829796344723593665420