L(s) = 1 | − 1.83·3-s + 7-s + 0.359·9-s + 4.40·11-s + 3.20·13-s − 1.14·17-s − 1.72·19-s − 1.83·21-s + 3.25·23-s + 4.83·27-s + 4.18·29-s + 1.36·31-s − 8.06·33-s − 4.19·37-s − 5.88·39-s + 11.7·41-s − 2.64·43-s + 0.106·47-s + 49-s + 2.10·51-s − 7.86·53-s + 3.16·57-s + 13.3·59-s − 10.0·61-s + 0.359·63-s + 1.28·67-s − 5.96·69-s + ⋯ |
L(s) = 1 | − 1.05·3-s + 0.377·7-s + 0.119·9-s + 1.32·11-s + 0.889·13-s − 0.278·17-s − 0.395·19-s − 0.399·21-s + 0.678·23-s + 0.931·27-s + 0.777·29-s + 0.245·31-s − 1.40·33-s − 0.690·37-s − 0.941·39-s + 1.83·41-s − 0.404·43-s + 0.0155·47-s + 0.142·49-s + 0.294·51-s − 1.08·53-s + 0.419·57-s + 1.73·59-s − 1.28·61-s + 0.0453·63-s + 0.156·67-s − 0.717·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.542604637\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.542604637\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 1.83T + 3T^{2} \) |
| 11 | \( 1 - 4.40T + 11T^{2} \) |
| 13 | \( 1 - 3.20T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 + 1.72T + 19T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 - 4.18T + 29T^{2} \) |
| 31 | \( 1 - 1.36T + 31T^{2} \) |
| 37 | \( 1 + 4.19T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 2.64T + 43T^{2} \) |
| 47 | \( 1 - 0.106T + 47T^{2} \) |
| 53 | \( 1 + 7.86T + 53T^{2} \) |
| 59 | \( 1 - 13.3T + 59T^{2} \) |
| 61 | \( 1 + 10.0T + 61T^{2} \) |
| 67 | \( 1 - 1.28T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 11.8T + 73T^{2} \) |
| 79 | \( 1 - 5.48T + 79T^{2} \) |
| 83 | \( 1 + 17.1T + 83T^{2} \) |
| 89 | \( 1 - 4.31T + 89T^{2} \) |
| 97 | \( 1 + 7.65T + 97T^{2} \) |
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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
−8.283994115145815512292113566851, −7.20944233214373357840736804319, −6.50717729387408318569690189377, −6.12504909026825829938338176991, −5.31518726839905365748484693949, −4.53844023141174070610623669582, −3.88496143879875476406589089924, −2.84162258515240097357752685420, −1.56735090245609113726275050633, −0.75535343310244052859004847512,
0.75535343310244052859004847512, 1.56735090245609113726275050633, 2.84162258515240097357752685420, 3.88496143879875476406589089924, 4.53844023141174070610623669582, 5.31518726839905365748484693949, 6.12504909026825829938338176991, 6.50717729387408318569690189377, 7.20944233214373357840736804319, 8.283994115145815512292113566851