L(s) = 1 | + 1.15·2-s − 0.220·3-s − 0.659·4-s − 0.421·5-s − 0.255·6-s − 0.645·7-s − 3.07·8-s − 2.95·9-s − 0.487·10-s − 0.640·11-s + 0.145·12-s − 4.07·13-s − 0.747·14-s + 0.0929·15-s − 2.24·16-s + 6.47·17-s − 3.41·18-s − 4.15·19-s + 0.277·20-s + 0.142·21-s − 0.741·22-s − 7.48·23-s + 0.679·24-s − 4.82·25-s − 4.71·26-s + 1.31·27-s + 0.425·28-s + ⋯ |
L(s) = 1 | + 0.818·2-s − 0.127·3-s − 0.329·4-s − 0.188·5-s − 0.104·6-s − 0.243·7-s − 1.08·8-s − 0.983·9-s − 0.154·10-s − 0.192·11-s + 0.0419·12-s − 1.12·13-s − 0.199·14-s + 0.0240·15-s − 0.561·16-s + 1.57·17-s − 0.805·18-s − 0.952·19-s + 0.0621·20-s + 0.0310·21-s − 0.158·22-s − 1.56·23-s + 0.138·24-s − 0.964·25-s − 0.924·26-s + 0.252·27-s + 0.0803·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 547 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 547 | \( 1 + T \) |
good | 2 | \( 1 - 1.15T + 2T^{2} \) |
| 3 | \( 1 + 0.220T + 3T^{2} \) |
| 5 | \( 1 + 0.421T + 5T^{2} \) |
| 7 | \( 1 + 0.645T + 7T^{2} \) |
| 11 | \( 1 + 0.640T + 11T^{2} \) |
| 13 | \( 1 + 4.07T + 13T^{2} \) |
| 17 | \( 1 - 6.47T + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 + 7.48T + 23T^{2} \) |
| 29 | \( 1 + 0.298T + 29T^{2} \) |
| 31 | \( 1 - 10.2T + 31T^{2} \) |
| 37 | \( 1 + 6.30T + 37T^{2} \) |
| 41 | \( 1 - 3.79T + 41T^{2} \) |
| 43 | \( 1 - 0.987T + 43T^{2} \) |
| 47 | \( 1 - 0.803T + 47T^{2} \) |
| 53 | \( 1 + 0.867T + 53T^{2} \) |
| 59 | \( 1 - 0.841T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 - 3.61T + 67T^{2} \) |
| 71 | \( 1 + 4.80T + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 - 5.23T + 79T^{2} \) |
| 83 | \( 1 + 17.0T + 83T^{2} \) |
| 89 | \( 1 + 9.06T + 89T^{2} \) |
| 97 | \( 1 - 5.13T + 97T^{2} \) |
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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
−10.23068804261339461236653121268, −9.651639831990040984906736503952, −8.455916343871390903655098621179, −7.77925729697143064995723929153, −6.30983924619455735658012761867, −5.63556699067308146067789074196, −4.69144464869815195290085995252, −3.62796897450870693902336149362, −2.57288271879330785701893290617, 0,
2.57288271879330785701893290617, 3.62796897450870693902336149362, 4.69144464869815195290085995252, 5.63556699067308146067789074196, 6.30983924619455735658012761867, 7.77925729697143064995723929153, 8.455916343871390903655098621179, 9.651639831990040984906736503952, 10.23068804261339461236653121268