L(s) = 1 | + 1.21·2-s − 0.534·4-s − 2.21·5-s − 3.06·8-s − 2.67·10-s + 0.789·11-s − 13-s − 2.64·16-s + 1.74·17-s + 4.32·19-s + 1.18·20-s + 0.955·22-s + 1.11·23-s − 0.112·25-s − 1.21·26-s + 8.48·29-s + 5.70·31-s + 2.93·32-s + 2.11·34-s + 2.27·37-s + 5.23·38-s + 6.78·40-s − 12.1·41-s + 8.06·43-s − 0.421·44-s + 1.34·46-s − 8.74·47-s + ⋯ |
L(s) = 1 | + 0.856·2-s − 0.267·4-s − 0.988·5-s − 1.08·8-s − 0.846·10-s + 0.237·11-s − 0.277·13-s − 0.661·16-s + 0.423·17-s + 0.991·19-s + 0.264·20-s + 0.203·22-s + 0.231·23-s − 0.0225·25-s − 0.237·26-s + 1.57·29-s + 1.02·31-s + 0.518·32-s + 0.362·34-s + 0.374·37-s + 0.849·38-s + 1.07·40-s − 1.89·41-s + 1.23·43-s − 0.0635·44-s + 0.198·46-s − 1.27·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5733 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 2 | \( 1 - 1.21T + 2T^{2} \) |
| 5 | \( 1 + 2.21T + 5T^{2} \) |
| 11 | \( 1 - 0.789T + 11T^{2} \) |
| 17 | \( 1 - 1.74T + 17T^{2} \) |
| 19 | \( 1 - 4.32T + 19T^{2} \) |
| 23 | \( 1 - 1.11T + 23T^{2} \) |
| 29 | \( 1 - 8.48T + 29T^{2} \) |
| 31 | \( 1 - 5.70T + 31T^{2} \) |
| 37 | \( 1 - 2.27T + 37T^{2} \) |
| 41 | \( 1 + 12.1T + 41T^{2} \) |
| 43 | \( 1 - 8.06T + 43T^{2} \) |
| 47 | \( 1 + 8.74T + 47T^{2} \) |
| 53 | \( 1 + 7.95T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 6.55T + 67T^{2} \) |
| 71 | \( 1 + 5.85T + 71T^{2} \) |
| 73 | \( 1 - 8.00T + 73T^{2} \) |
| 79 | \( 1 + 6.91T + 79T^{2} \) |
| 83 | \( 1 + 3.14T + 83T^{2} \) |
| 89 | \( 1 + 3.39T + 89T^{2} \) |
| 97 | \( 1 + 0.0981T + 97T^{2} \) |
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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
−7.956783821011992985610054765422, −6.90839018769688655683602901641, −6.29734938509435267312377343293, −5.39734928446845248975801914006, −4.71130323537766228405095667017, −4.21554498710043364305299175627, −3.27350512316635912566914136701, −2.88348119257592294393866809695, −1.23855841890560397048925522725, 0,
1.23855841890560397048925522725, 2.88348119257592294393866809695, 3.27350512316635912566914136701, 4.21554498710043364305299175627, 4.71130323537766228405095667017, 5.39734928446845248975801914006, 6.29734938509435267312377343293, 6.90839018769688655683602901641, 7.956783821011992985610054765422