L(s) = 1 | + 6·11-s − 13-s + 2·17-s + 4·23-s − 6·29-s − 4·31-s − 2·37-s + 4·43-s + 10·47-s − 7·49-s + 10·53-s − 6·59-s + 6·61-s − 12·67-s − 2·71-s − 6·73-s − 16·79-s − 6·83-s − 4·89-s − 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
L(s) = 1 | + 1.80·11-s − 0.277·13-s + 0.485·17-s + 0.834·23-s − 1.11·29-s − 0.718·31-s − 0.328·37-s + 0.609·43-s + 1.45·47-s − 49-s + 1.37·53-s − 0.781·59-s + 0.768·61-s − 1.46·67-s − 0.237·71-s − 0.702·73-s − 1.80·79-s − 0.658·83-s − 0.423·89-s − 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 187200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.698070662\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.698070662\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−13.15630669548725, −12.55111060980002, −12.20099070848266, −11.70578938634921, −11.31704594829010, −10.84955285956033, −10.28855605830723, −9.701088500319142, −9.334070618446349, −8.818679260625095, −8.611834621987324, −7.722642298115513, −7.194651037875845, −7.040447519227399, −6.335865287623956, −5.695158588555209, −5.520742775690209, −4.595710646742709, −4.193173129251858, −3.693893784860427, −3.140176340853963, −2.494926474300258, −1.635409117865603, −1.321412572376089, −0.4819229395445539,
0.4819229395445539, 1.321412572376089, 1.635409117865603, 2.494926474300258, 3.140176340853963, 3.693893784860427, 4.193173129251858, 4.595710646742709, 5.520742775690209, 5.695158588555209, 6.335865287623956, 7.040447519227399, 7.194651037875845, 7.722642298115513, 8.611834621987324, 8.818679260625095, 9.334070618446349, 9.701088500319142, 10.28855605830723, 10.84955285956033, 11.31704594829010, 11.70578938634921, 12.20099070848266, 12.55111060980002, 13.15630669548725