| L(s) = 1 | − 2.57·2-s + 1.72·3-s − 1.37·4-s + 7.29·5-s − 4.44·6-s + 28.8·7-s + 24.1·8-s − 24.0·9-s − 18.7·10-s + 14.0·11-s − 2.37·12-s − 51.2·13-s − 74.1·14-s + 12.5·15-s − 51.1·16-s − 33.1·17-s + 61.8·18-s − 73.1·19-s − 10.0·20-s + 49.7·21-s − 36.0·22-s − 23·23-s + 41.6·24-s − 71.8·25-s + 132.·26-s − 88.0·27-s − 39.5·28-s + ⋯ |
| L(s) = 1 | − 0.910·2-s + 0.332·3-s − 0.171·4-s + 0.652·5-s − 0.302·6-s + 1.55·7-s + 1.06·8-s − 0.889·9-s − 0.593·10-s + 0.383·11-s − 0.0570·12-s − 1.09·13-s − 1.41·14-s + 0.216·15-s − 0.798·16-s − 0.473·17-s + 0.809·18-s − 0.882·19-s − 0.111·20-s + 0.517·21-s − 0.349·22-s − 0.208·23-s + 0.354·24-s − 0.574·25-s + 0.996·26-s − 0.627·27-s − 0.267·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 667 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 + 23T \) |
| 29 | \( 1 - 29T \) |
| good | 2 | \( 1 + 2.57T + 8T^{2} \) |
| 3 | \( 1 - 1.72T + 27T^{2} \) |
| 5 | \( 1 - 7.29T + 125T^{2} \) |
| 7 | \( 1 - 28.8T + 343T^{2} \) |
| 11 | \( 1 - 14.0T + 1.33e3T^{2} \) |
| 13 | \( 1 + 51.2T + 2.19e3T^{2} \) |
| 17 | \( 1 + 33.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 73.1T + 6.85e3T^{2} \) |
| 31 | \( 1 + 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 43.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 340.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 328.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 17.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 680.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 9.64T + 2.05e5T^{2} \) |
| 61 | \( 1 + 918.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 561.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 483.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 94.6T + 3.89e5T^{2} \) |
| 79 | \( 1 + 583.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.14e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.06e3T + 9.12e5T^{2} \) |
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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
−9.271037882407603904037726703917, −9.065185941704515033792447809529, −7.968462223559171023842079665801, −7.57577921188118617765889625114, −6.08463638645164364543009195766, −5.02699012776624793026300181306, −4.22791759278643648812451073813, −2.38005989069093757114890124655, −1.58821867454364133450026276732, 0,
1.58821867454364133450026276732, 2.38005989069093757114890124655, 4.22791759278643648812451073813, 5.02699012776624793026300181306, 6.08463638645164364543009195766, 7.57577921188118617765889625114, 7.968462223559171023842079665801, 9.065185941704515033792447809529, 9.271037882407603904037726703917
