L(s) = 1 | − 0.454·5-s − 2.18·7-s + 1.15·11-s + 6.03·13-s − 0.0535·17-s − 2.08·19-s − 3.22·23-s − 4.79·25-s + 1.62·29-s − 2.01·31-s + 0.993·35-s − 3.49·37-s − 2.85·41-s + 2.78·43-s + 9.03·47-s − 2.21·49-s + 4.09·53-s − 0.526·55-s + 4.76·59-s − 14.6·61-s − 2.74·65-s − 4.98·67-s + 2.04·71-s − 0.0937·73-s − 2.53·77-s − 1.92·79-s − 7.96·83-s + ⋯ |
L(s) = 1 | − 0.203·5-s − 0.826·7-s + 0.349·11-s + 1.67·13-s − 0.0129·17-s − 0.479·19-s − 0.673·23-s − 0.958·25-s + 0.300·29-s − 0.362·31-s + 0.167·35-s − 0.574·37-s − 0.446·41-s + 0.423·43-s + 1.31·47-s − 0.317·49-s + 0.561·53-s − 0.0710·55-s + 0.620·59-s − 1.87·61-s − 0.340·65-s − 0.609·67-s + 0.242·71-s − 0.0109·73-s − 0.288·77-s − 0.216·79-s − 0.874·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9036 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 + T \) |
good | 5 | \( 1 + 0.454T + 5T^{2} \) |
| 7 | \( 1 + 2.18T + 7T^{2} \) |
| 11 | \( 1 - 1.15T + 11T^{2} \) |
| 13 | \( 1 - 6.03T + 13T^{2} \) |
| 17 | \( 1 + 0.0535T + 17T^{2} \) |
| 19 | \( 1 + 2.08T + 19T^{2} \) |
| 23 | \( 1 + 3.22T + 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 2.01T + 31T^{2} \) |
| 37 | \( 1 + 3.49T + 37T^{2} \) |
| 41 | \( 1 + 2.85T + 41T^{2} \) |
| 43 | \( 1 - 2.78T + 43T^{2} \) |
| 47 | \( 1 - 9.03T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 + 14.6T + 61T^{2} \) |
| 67 | \( 1 + 4.98T + 67T^{2} \) |
| 71 | \( 1 - 2.04T + 71T^{2} \) |
| 73 | \( 1 + 0.0937T + 73T^{2} \) |
| 79 | \( 1 + 1.92T + 79T^{2} \) |
| 83 | \( 1 + 7.96T + 83T^{2} \) |
| 89 | \( 1 - 16.7T + 89T^{2} \) |
| 97 | \( 1 - 3.00T + 97T^{2} \) |
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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
−7.39998199104708574258525399949, −6.54269970359087954925261831863, −6.12915897333396176329049174668, −5.53193482275613286110424105139, −4.37722237699094667308227741076, −3.79649144760933204555594517155, −3.25360386900404939876376984563, −2.16668427083624228992803294578, −1.21095353457914458271724858244, 0,
1.21095353457914458271724858244, 2.16668427083624228992803294578, 3.25360386900404939876376984563, 3.79649144760933204555594517155, 4.37722237699094667308227741076, 5.53193482275613286110424105139, 6.12915897333396176329049174668, 6.54269970359087954925261831863, 7.39998199104708574258525399949