| L(s) = 1 | − 2-s + 4-s − 2.69·5-s + 2.17·7-s − 8-s + 2.69·10-s + 2.79·11-s + 5.60·13-s − 2.17·14-s + 16-s + 4.05·17-s + 3.43·19-s − 2.69·20-s − 2.79·22-s + 0.907·23-s + 2.24·25-s − 5.60·26-s + 2.17·28-s − 3.53·29-s + 2.49·31-s − 32-s − 4.05·34-s − 5.85·35-s + 4.44·37-s − 3.43·38-s + 2.69·40-s + 10.8·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.20·5-s + 0.821·7-s − 0.353·8-s + 0.851·10-s + 0.844·11-s + 1.55·13-s − 0.581·14-s + 0.250·16-s + 0.982·17-s + 0.789·19-s − 0.601·20-s − 0.596·22-s + 0.189·23-s + 0.448·25-s − 1.09·26-s + 0.410·28-s − 0.656·29-s + 0.448·31-s − 0.176·32-s − 0.694·34-s − 0.989·35-s + 0.730·37-s − 0.558·38-s + 0.425·40-s + 1.70·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.485123560\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.485123560\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 223 | \( 1 + T \) |
| good | 5 | \( 1 + 2.69T + 5T^{2} \) |
| 7 | \( 1 - 2.17T + 7T^{2} \) |
| 11 | \( 1 - 2.79T + 11T^{2} \) |
| 13 | \( 1 - 5.60T + 13T^{2} \) |
| 17 | \( 1 - 4.05T + 17T^{2} \) |
| 19 | \( 1 - 3.43T + 19T^{2} \) |
| 23 | \( 1 - 0.907T + 23T^{2} \) |
| 29 | \( 1 + 3.53T + 29T^{2} \) |
| 31 | \( 1 - 2.49T + 31T^{2} \) |
| 37 | \( 1 - 4.44T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 7.91T + 43T^{2} \) |
| 47 | \( 1 - 13.4T + 47T^{2} \) |
| 53 | \( 1 + 13.1T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 0.466T + 61T^{2} \) |
| 67 | \( 1 - 6.52T + 67T^{2} \) |
| 71 | \( 1 + 2.84T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 + 7.97T + 79T^{2} \) |
| 83 | \( 1 - 1.80T + 83T^{2} \) |
| 89 | \( 1 + 10.9T + 89T^{2} \) |
| 97 | \( 1 - 16.3T + 97T^{2} \) |
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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
−8.279718414282224779870842486404, −7.85939653476212534257837548600, −7.30730203779238170624042727932, −6.32432432992132587556521186764, −5.63217216319776060335251484588, −4.49547922795136965110948220853, −3.77904802022638869271309710644, −3.07499336586676669807746869374, −1.56034687897262026486875557278, −0.866666945853861052927049627336,
0.866666945853861052927049627336, 1.56034687897262026486875557278, 3.07499336586676669807746869374, 3.77904802022638869271309710644, 4.49547922795136965110948220853, 5.63217216319776060335251484588, 6.32432432992132587556521186764, 7.30730203779238170624042727932, 7.85939653476212534257837548600, 8.279718414282224779870842486404
