| L(s) = 1 | − 3·3-s − 8·4-s − 12·5-s + 2·7-s + 9·9-s − 36·11-s + 24·12-s + 13·13-s + 36·15-s + 64·16-s − 78·17-s + 74·19-s + 96·20-s − 6·21-s − 96·23-s + 19·25-s − 27·27-s − 16·28-s + 18·29-s − 214·31-s + 108·33-s − 24·35-s − 72·36-s − 286·37-s − 39·39-s − 384·41-s + 524·43-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 1.07·5-s + 0.107·7-s + 1/3·9-s − 0.986·11-s + 0.577·12-s + 0.277·13-s + 0.619·15-s + 16-s − 1.11·17-s + 0.893·19-s + 1.07·20-s − 0.0623·21-s − 0.870·23-s + 0.151·25-s − 0.192·27-s − 0.107·28-s + 0.115·29-s − 1.23·31-s + 0.569·33-s − 0.115·35-s − 1/3·36-s − 1.27·37-s − 0.160·39-s − 1.46·41-s + 1.85·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + p T \) |
| 13 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 7 | \( 1 - 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 36 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 - 74 T + p^{3} T^{2} \) |
| 23 | \( 1 + 96 T + p^{3} T^{2} \) |
| 29 | \( 1 - 18 T + p^{3} T^{2} \) |
| 31 | \( 1 + 214 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 + 384 T + p^{3} T^{2} \) |
| 43 | \( 1 - 524 T + p^{3} T^{2} \) |
| 47 | \( 1 - 300 T + p^{3} T^{2} \) |
| 53 | \( 1 - 558 T + p^{3} T^{2} \) |
| 59 | \( 1 - 576 T + p^{3} T^{2} \) |
| 61 | \( 1 - 74 T + p^{3} T^{2} \) |
| 67 | \( 1 - 38 T + p^{3} T^{2} \) |
| 71 | \( 1 + 456 T + p^{3} T^{2} \) |
| 73 | \( 1 + 682 T + p^{3} T^{2} \) |
| 79 | \( 1 - 704 T + p^{3} T^{2} \) |
| 83 | \( 1 + 888 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1020 T + p^{3} T^{2} \) |
| 97 | \( 1 - 110 T + p^{3} T^{2} \) |
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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
−15.37339423930603975939502126622, −13.84856769429132218900969126442, −12.73492362541996679315999732284, −11.56631585492723582184890274840, −10.30963685405775737810975281411, −8.706258406048468557221625959852, −7.46910472765699429393603148911, −5.37354050660650677480013717173, −3.98153049779614992197458007456, 0,
3.98153049779614992197458007456, 5.37354050660650677480013717173, 7.46910472765699429393603148911, 8.706258406048468557221625959852, 10.30963685405775737810975281411, 11.56631585492723582184890274840, 12.73492362541996679315999732284, 13.84856769429132218900969126442, 15.37339423930603975939502126622
