L(s) = 1 | + 0.0822·3-s − 4.16·5-s − 7-s − 2.99·9-s − 11-s − 13-s − 0.342·15-s + 3.55·17-s + 5.09·19-s − 0.0822·21-s − 0.867·23-s + 12.3·25-s − 0.492·27-s − 8.23·29-s + 2.28·31-s − 0.0822·33-s + 4.16·35-s + 6.74·37-s − 0.0822·39-s + 2.15·41-s + 6.41·43-s + 12.4·45-s − 13.3·47-s + 49-s + 0.292·51-s − 2.36·53-s + 4.16·55-s + ⋯ |
L(s) = 1 | + 0.0474·3-s − 1.86·5-s − 0.377·7-s − 0.997·9-s − 0.301·11-s − 0.277·13-s − 0.0884·15-s + 0.863·17-s + 1.16·19-s − 0.0179·21-s − 0.180·23-s + 2.47·25-s − 0.0948·27-s − 1.52·29-s + 0.410·31-s − 0.0143·33-s + 0.704·35-s + 1.10·37-s − 0.0131·39-s + 0.336·41-s + 0.978·43-s + 1.85·45-s − 1.94·47-s + 0.142·49-s + 0.0409·51-s − 0.324·53-s + 0.561·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
good | 3 | \( 1 - 0.0822T + 3T^{2} \) |
| 5 | \( 1 + 4.16T + 5T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 19 | \( 1 - 5.09T + 19T^{2} \) |
| 23 | \( 1 + 0.867T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 2.28T + 31T^{2} \) |
| 37 | \( 1 - 6.74T + 37T^{2} \) |
| 41 | \( 1 - 2.15T + 41T^{2} \) |
| 43 | \( 1 - 6.41T + 43T^{2} \) |
| 47 | \( 1 + 13.3T + 47T^{2} \) |
| 53 | \( 1 + 2.36T + 53T^{2} \) |
| 59 | \( 1 - 2.85T + 59T^{2} \) |
| 61 | \( 1 - 3.08T + 61T^{2} \) |
| 67 | \( 1 - 7.23T + 67T^{2} \) |
| 71 | \( 1 - 1.56T + 71T^{2} \) |
| 73 | \( 1 - 4.34T + 73T^{2} \) |
| 79 | \( 1 - 0.0822T + 79T^{2} \) |
| 83 | \( 1 + 5.58T + 83T^{2} \) |
| 89 | \( 1 - 5.43T + 89T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−7.72618694905280351932706839884, −7.02684571013490978346859200870, −6.07476586076651330305426925491, −5.33492430135165461647354519577, −4.63780527731195108694955509466, −3.65270290589470474259906183677, −3.33410815836188963593938507222, −2.50986716267567213031602799535, −0.914648945021387609450356470762, 0,
0.914648945021387609450356470762, 2.50986716267567213031602799535, 3.33410815836188963593938507222, 3.65270290589470474259906183677, 4.63780527731195108694955509466, 5.33492430135165461647354519577, 6.07476586076651330305426925491, 7.02684571013490978346859200870, 7.72618694905280351932706839884