L(s) = 1 | + 1.18·3-s + 0.677·5-s + 1.46·7-s − 1.58·9-s − 2.15·11-s + 4.65·13-s + 0.805·15-s − 17-s − 6.39·19-s + 1.73·21-s − 8.06·23-s − 4.54·25-s − 5.45·27-s − 0.503·29-s − 6.46·31-s − 2.56·33-s + 0.990·35-s − 4.81·37-s + 5.53·39-s + 3.96·41-s − 3.76·43-s − 1.07·45-s − 5.61·47-s − 4.86·49-s − 1.18·51-s + 5.58·53-s − 1.46·55-s + ⋯ |
L(s) = 1 | + 0.686·3-s + 0.303·5-s + 0.552·7-s − 0.529·9-s − 0.650·11-s + 1.29·13-s + 0.207·15-s − 0.242·17-s − 1.46·19-s + 0.379·21-s − 1.68·23-s − 0.908·25-s − 1.04·27-s − 0.0934·29-s − 1.16·31-s − 0.446·33-s + 0.167·35-s − 0.791·37-s + 0.886·39-s + 0.618·41-s − 0.573·43-s − 0.160·45-s − 0.818·47-s − 0.694·49-s − 0.166·51-s + 0.767·53-s − 0.197·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4012 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 17 | \( 1 + T \) |
| 59 | \( 1 - T \) |
good | 3 | \( 1 - 1.18T + 3T^{2} \) |
| 5 | \( 1 - 0.677T + 5T^{2} \) |
| 7 | \( 1 - 1.46T + 7T^{2} \) |
| 11 | \( 1 + 2.15T + 11T^{2} \) |
| 13 | \( 1 - 4.65T + 13T^{2} \) |
| 19 | \( 1 + 6.39T + 19T^{2} \) |
| 23 | \( 1 + 8.06T + 23T^{2} \) |
| 29 | \( 1 + 0.503T + 29T^{2} \) |
| 31 | \( 1 + 6.46T + 31T^{2} \) |
| 37 | \( 1 + 4.81T + 37T^{2} \) |
| 41 | \( 1 - 3.96T + 41T^{2} \) |
| 43 | \( 1 + 3.76T + 43T^{2} \) |
| 47 | \( 1 + 5.61T + 47T^{2} \) |
| 53 | \( 1 - 5.58T + 53T^{2} \) |
| 61 | \( 1 + 2.84T + 61T^{2} \) |
| 67 | \( 1 - 3.63T + 67T^{2} \) |
| 71 | \( 1 - 6.38T + 71T^{2} \) |
| 73 | \( 1 - 12.0T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 1.34T + 83T^{2} \) |
| 89 | \( 1 - 18.7T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.213551354584324434017323561301, −7.65461114342850483389765683276, −6.45826207189348241013924864654, −5.92393846735680841162233234502, −5.14942257567053977176850900243, −4.03983670028627472369336426224, −3.50643754860704849601225970021, −2.25723290941815816973851536964, −1.81665253346831263061175210437, 0,
1.81665253346831263061175210437, 2.25723290941815816973851536964, 3.50643754860704849601225970021, 4.03983670028627472369336426224, 5.14942257567053977176850900243, 5.92393846735680841162233234502, 6.45826207189348241013924864654, 7.65461114342850483389765683276, 8.213551354584324434017323561301