L(s) = 1 | + 3-s − 2.41·7-s + 9-s + 0.414·11-s + 13-s − 3.82·17-s − 2·19-s − 2.41·21-s + 0.828·23-s + 27-s − 8.65·29-s + 4.41·31-s + 0.414·33-s + 7.65·37-s + 39-s + 5.65·41-s + 10.4·43-s + 2.07·47-s − 1.17·49-s − 3.82·51-s + 5.82·53-s − 2·57-s − 6.41·59-s − 1.82·61-s − 2.41·63-s − 0.757·67-s + 0.828·69-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.912·7-s + 0.333·9-s + 0.124·11-s + 0.277·13-s − 0.928·17-s − 0.458·19-s − 0.526·21-s + 0.172·23-s + 0.192·27-s − 1.60·29-s + 0.792·31-s + 0.0721·33-s + 1.25·37-s + 0.160·39-s + 0.883·41-s + 1.59·43-s + 0.302·47-s − 0.167·49-s − 0.536·51-s + 0.800·53-s − 0.264·57-s − 0.835·59-s − 0.234·61-s − 0.304·63-s − 0.0925·67-s + 0.0997·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 2.41T + 7T^{2} \) |
| 11 | \( 1 - 0.414T + 11T^{2} \) |
| 17 | \( 1 + 3.82T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 + 8.65T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 5.65T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 2.07T + 47T^{2} \) |
| 53 | \( 1 - 5.82T + 53T^{2} \) |
| 59 | \( 1 + 6.41T + 59T^{2} \) |
| 61 | \( 1 + 1.82T + 61T^{2} \) |
| 67 | \( 1 + 0.757T + 67T^{2} \) |
| 71 | \( 1 + 7.65T + 71T^{2} \) |
| 73 | \( 1 + 9.31T + 73T^{2} \) |
| 79 | \( 1 + 8.82T + 79T^{2} \) |
| 83 | \( 1 + 13.7T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 14.9T + 97T^{2} \) |
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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
−7.49662425583364825893154870058, −6.86861963198206200584581146741, −6.15197511404963312029320639393, −5.59761566725366290860496508345, −4.28478198181978196413609115378, −4.08813232584514651253703421161, −2.95835306723105811582039678931, −2.46410789823083094832853069126, −1.32667992047723484410723322672, 0,
1.32667992047723484410723322672, 2.46410789823083094832853069126, 2.95835306723105811582039678931, 4.08813232584514651253703421161, 4.28478198181978196413609115378, 5.59761566725366290860496508345, 6.15197511404963312029320639393, 6.86861963198206200584581146741, 7.49662425583364825893154870058