| L(s) = 1 | + 3-s + 0.273·5-s − 2.92·7-s + 9-s − 4.02·11-s + 1.67·13-s + 0.273·15-s + 0.0598·17-s + 5.15·19-s − 2.92·21-s + 0.580·23-s − 4.92·25-s + 27-s + 1.19·29-s + 3.68·31-s − 4.02·33-s − 0.800·35-s + 6.70·37-s + 1.67·39-s − 7.90·41-s + 7.84·43-s + 0.273·45-s − 9.18·47-s + 1.57·49-s + 0.0598·51-s + 7.76·53-s − 1.09·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.122·5-s − 1.10·7-s + 0.333·9-s − 1.21·11-s + 0.465·13-s + 0.0705·15-s + 0.0145·17-s + 1.18·19-s − 0.638·21-s + 0.120·23-s − 0.985·25-s + 0.192·27-s + 0.221·29-s + 0.661·31-s − 0.699·33-s − 0.135·35-s + 1.10·37-s + 0.268·39-s − 1.23·41-s + 1.19·43-s + 0.0407·45-s − 1.34·47-s + 0.224·49-s + 0.00837·51-s + 1.06·53-s − 0.148·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.930636487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930636487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 251 | \( 1 + T \) |
| good | 5 | \( 1 - 0.273T + 5T^{2} \) |
| 7 | \( 1 + 2.92T + 7T^{2} \) |
| 11 | \( 1 + 4.02T + 11T^{2} \) |
| 13 | \( 1 - 1.67T + 13T^{2} \) |
| 17 | \( 1 - 0.0598T + 17T^{2} \) |
| 19 | \( 1 - 5.15T + 19T^{2} \) |
| 23 | \( 1 - 0.580T + 23T^{2} \) |
| 29 | \( 1 - 1.19T + 29T^{2} \) |
| 31 | \( 1 - 3.68T + 31T^{2} \) |
| 37 | \( 1 - 6.70T + 37T^{2} \) |
| 41 | \( 1 + 7.90T + 41T^{2} \) |
| 43 | \( 1 - 7.84T + 43T^{2} \) |
| 47 | \( 1 + 9.18T + 47T^{2} \) |
| 53 | \( 1 - 7.76T + 53T^{2} \) |
| 59 | \( 1 + 1.33T + 59T^{2} \) |
| 61 | \( 1 - 4.78T + 61T^{2} \) |
| 67 | \( 1 + 5.67T + 67T^{2} \) |
| 71 | \( 1 + 5.64T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 + 14.1T + 79T^{2} \) |
| 83 | \( 1 - 14.6T + 83T^{2} \) |
| 89 | \( 1 - 8.55T + 89T^{2} \) |
| 97 | \( 1 + 0.970T + 97T^{2} \) |
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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
−7.991038963434644738222837944983, −7.51888190292160401106896080924, −6.67563899871162605079732709614, −5.97457228953234758223423265119, −5.27737735217686738903397538771, −4.35992315871557937000200227747, −3.37392904158523882586654934451, −2.95191814498684034974991577436, −2.02328824913486073537344470007, −0.69084006526072261046098045117,
0.69084006526072261046098045117, 2.02328824913486073537344470007, 2.95191814498684034974991577436, 3.37392904158523882586654934451, 4.35992315871557937000200227747, 5.27737735217686738903397538771, 5.97457228953234758223423265119, 6.67563899871162605079732709614, 7.51888190292160401106896080924, 7.991038963434644738222837944983
