L(s) = 1 | − 3-s − 4.42·5-s − 1.88·7-s + 9-s − 1.84·11-s − 5.16·13-s + 4.42·15-s + 0.0891·17-s − 6.45·19-s + 1.88·21-s + 3.21·23-s + 14.6·25-s − 27-s + 3.02·29-s + 10.4·31-s + 1.84·33-s + 8.34·35-s − 1.51·37-s + 5.16·39-s − 7.82·41-s + 7.61·43-s − 4.42·45-s + 11.5·47-s − 3.45·49-s − 0.0891·51-s − 2.84·53-s + 8.17·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 1.98·5-s − 0.712·7-s + 0.333·9-s − 0.556·11-s − 1.43·13-s + 1.14·15-s + 0.0216·17-s − 1.48·19-s + 0.411·21-s + 0.669·23-s + 2.92·25-s − 0.192·27-s + 0.560·29-s + 1.87·31-s + 0.321·33-s + 1.41·35-s − 0.248·37-s + 0.827·39-s − 1.22·41-s + 1.16·43-s − 0.660·45-s + 1.68·47-s − 0.492·49-s − 0.0124·51-s − 0.390·53-s + 1.10·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 167 | \( 1 - T \) |
good | 5 | \( 1 + 4.42T + 5T^{2} \) |
| 7 | \( 1 + 1.88T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 + 5.16T + 13T^{2} \) |
| 17 | \( 1 - 0.0891T + 17T^{2} \) |
| 19 | \( 1 + 6.45T + 19T^{2} \) |
| 23 | \( 1 - 3.21T + 23T^{2} \) |
| 29 | \( 1 - 3.02T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 1.51T + 37T^{2} \) |
| 41 | \( 1 + 7.82T + 41T^{2} \) |
| 43 | \( 1 - 7.61T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 2.84T + 53T^{2} \) |
| 59 | \( 1 - 5.92T + 59T^{2} \) |
| 61 | \( 1 - 4.29T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 - 11.9T + 71T^{2} \) |
| 73 | \( 1 + 0.174T + 73T^{2} \) |
| 79 | \( 1 + 2.45T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 + 18.3T + 89T^{2} \) |
| 97 | \( 1 + 3.13T + 97T^{2} \) |
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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
−7.34402886141487484221350417606, −6.95075844071126569459657019491, −6.29923847020469100834372137236, −5.18013498762243910272611584163, −4.56518231863875818469727879006, −4.10440720674594068398536264888, −3.11372983172296653539961160481, −2.49357388335650402604627006119, −0.74222729923265057508952492241, 0,
0.74222729923265057508952492241, 2.49357388335650402604627006119, 3.11372983172296653539961160481, 4.10440720674594068398536264888, 4.56518231863875818469727879006, 5.18013498762243910272611584163, 6.29923847020469100834372137236, 6.95075844071126569459657019491, 7.34402886141487484221350417606