L(s) = 1 | + 2.87·3-s − 3.46·5-s − 0.332·7-s + 5.24·9-s − 2.63·11-s − 1.68·13-s − 9.95·15-s + 7.27·17-s − 0.928·19-s − 0.955·21-s − 6.01·23-s + 7.02·25-s + 6.44·27-s − 6.02·29-s + 1.65·31-s − 7.57·33-s + 1.15·35-s + 2.35·37-s − 4.84·39-s + 3.70·41-s − 11.4·43-s − 18.1·45-s − 2.14·47-s − 6.88·49-s + 20.8·51-s + 4.74·53-s + 9.15·55-s + ⋯ |
L(s) = 1 | + 1.65·3-s − 1.55·5-s − 0.125·7-s + 1.74·9-s − 0.795·11-s − 0.467·13-s − 2.57·15-s + 1.76·17-s − 0.213·19-s − 0.208·21-s − 1.25·23-s + 1.40·25-s + 1.23·27-s − 1.11·29-s + 0.297·31-s − 1.31·33-s + 0.195·35-s + 0.387·37-s − 0.775·39-s + 0.579·41-s − 1.74·43-s − 2.71·45-s − 0.312·47-s − 0.984·49-s + 2.92·51-s + 0.651·53-s + 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
good | 3 | \( 1 - 2.87T + 3T^{2} \) |
| 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 0.332T + 7T^{2} \) |
| 11 | \( 1 + 2.63T + 11T^{2} \) |
| 13 | \( 1 + 1.68T + 13T^{2} \) |
| 17 | \( 1 - 7.27T + 17T^{2} \) |
| 19 | \( 1 + 0.928T + 19T^{2} \) |
| 23 | \( 1 + 6.01T + 23T^{2} \) |
| 29 | \( 1 + 6.02T + 29T^{2} \) |
| 31 | \( 1 - 1.65T + 31T^{2} \) |
| 37 | \( 1 - 2.35T + 37T^{2} \) |
| 41 | \( 1 - 3.70T + 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 + 2.14T + 47T^{2} \) |
| 53 | \( 1 - 4.74T + 53T^{2} \) |
| 59 | \( 1 + 8.11T + 59T^{2} \) |
| 61 | \( 1 + 12.6T + 61T^{2} \) |
| 67 | \( 1 - 4.45T + 67T^{2} \) |
| 71 | \( 1 - 8.96T + 71T^{2} \) |
| 73 | \( 1 + 8.01T + 73T^{2} \) |
| 79 | \( 1 + 16.0T + 79T^{2} \) |
| 83 | \( 1 + 15.6T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 + 4.15T + 97T^{2} \) |
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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
−8.099818817093831606073942295639, −7.62291829544130686370739815727, −7.13313603993109557321953730371, −5.83902671322524560335813028747, −4.75937255632728757726065199049, −3.97654176814884240616066641238, −3.34789904385968693901449560346, −2.79617869580730157934546795001, −1.63243698416375054402107691020, 0,
1.63243698416375054402107691020, 2.79617869580730157934546795001, 3.34789904385968693901449560346, 3.97654176814884240616066641238, 4.75937255632728757726065199049, 5.83902671322524560335813028747, 7.13313603993109557321953730371, 7.62291829544130686370739815727, 8.099818817093831606073942295639