| L(s) = 1 | − 3-s + 9-s − 6·11-s − 4·13-s − 3·17-s − 4·19-s + 3·23-s − 27-s − 6·29-s + 5·31-s + 6·33-s − 8·37-s + 4·39-s + 3·41-s + 8·43-s − 9·47-s + 3·51-s − 12·53-s + 4·57-s + 6·59-s − 2·61-s + 8·67-s − 3·69-s + 9·71-s + 14·73-s + 7·79-s + 81-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 1.80·11-s − 1.10·13-s − 0.727·17-s − 0.917·19-s + 0.625·23-s − 0.192·27-s − 1.11·29-s + 0.898·31-s + 1.04·33-s − 1.31·37-s + 0.640·39-s + 0.468·41-s + 1.21·43-s − 1.31·47-s + 0.420·51-s − 1.64·53-s + 0.529·57-s + 0.781·59-s − 0.256·61-s + 0.977·67-s − 0.361·69-s + 1.06·71-s + 1.63·73-s + 0.787·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 7 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−14.55078087164620, −14.15951476142324, −13.35109940602199, −13.00886384799779, −12.61322311408347, −12.20450718459804, −11.39510114505876, −10.96857855110632, −10.62149787427790, −10.07793241786489, −9.520417475093207, −9.017922605100465, −8.138332378201710, −7.915698619643814, −7.243839073220488, −6.699743685002921, −6.190350914084797, −5.370574889249014, −5.027144332193383, −4.635486434160748, −3.807072159767105, −3.019476567196474, −2.312997891436848, −1.944625360141018, −0.6420638674460017, 0,
0.6420638674460017, 1.944625360141018, 2.312997891436848, 3.019476567196474, 3.807072159767105, 4.635486434160748, 5.027144332193383, 5.370574889249014, 6.190350914084797, 6.699743685002921, 7.243839073220488, 7.915698619643814, 8.138332378201710, 9.017922605100465, 9.520417475093207, 10.07793241786489, 10.62149787427790, 10.96857855110632, 11.39510114505876, 12.20450718459804, 12.61322311408347, 13.00886384799779, 13.35109940602199, 14.15951476142324, 14.55078087164620
