| L(s) = 1 | − 2-s + 0.302·3-s + 4-s − 1.30·5-s − 0.302·6-s − 8-s − 2.90·9-s + 1.30·10-s + 1.30·11-s + 0.302·12-s + 2.30·13-s − 0.394·15-s + 16-s + 6·17-s + 2.90·18-s − 2·19-s − 1.30·20-s − 1.30·22-s − 6.90·23-s − 0.302·24-s − 3.30·25-s − 2.30·26-s − 1.78·27-s + 6.90·29-s + 0.394·30-s − 3.30·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.174·3-s + 0.5·4-s − 0.582·5-s − 0.123·6-s − 0.353·8-s − 0.969·9-s + 0.411·10-s + 0.392·11-s + 0.0874·12-s + 0.638·13-s − 0.101·15-s + 0.250·16-s + 1.45·17-s + 0.685·18-s − 0.458·19-s − 0.291·20-s − 0.277·22-s − 1.44·23-s − 0.0618·24-s − 0.660·25-s − 0.451·26-s − 0.344·27-s + 1.28·29-s + 0.0720·30-s − 0.593·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3626 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3626 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 0.302T + 3T^{2} \) |
| 5 | \( 1 + 1.30T + 5T^{2} \) |
| 11 | \( 1 - 1.30T + 11T^{2} \) |
| 13 | \( 1 - 2.30T + 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 6.90T + 23T^{2} \) |
| 29 | \( 1 - 6.90T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 41 | \( 1 - 0.908T + 41T^{2} \) |
| 43 | \( 1 + 6.60T + 43T^{2} \) |
| 47 | \( 1 - 2.60T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 3.39T + 59T^{2} \) |
| 61 | \( 1 - 10.5T + 61T^{2} \) |
| 67 | \( 1 - 14.5T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 - 8.69T + 73T^{2} \) |
| 79 | \( 1 + 16.1T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 + 5.21T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−8.250702105335934015549438462891, −7.76076160882604891676030583576, −6.76054913774309270867412444465, −6.03000361971879034146966875015, −5.36308393485231880873654947257, −4.04248073808316820144615592023, −3.44414620999685219511328705165, −2.45722966882936373363677010627, −1.29516347236774972406017734113, 0,
1.29516347236774972406017734113, 2.45722966882936373363677010627, 3.44414620999685219511328705165, 4.04248073808316820144615592023, 5.36308393485231880873654947257, 6.03000361971879034146966875015, 6.76054913774309270867412444465, 7.76076160882604891676030583576, 8.250702105335934015549438462891
