L(s) = 1 | − 2-s + 4-s + 4.76·7-s − 8-s − 0.960·11-s + 2.24·13-s − 4.76·14-s + 16-s + 0.249·17-s + 19-s + 0.960·22-s + 9.01·23-s − 2.24·26-s + 4.76·28-s − 6.24·29-s + 2.96·31-s − 32-s − 0.249·34-s − 0.0399·37-s − 38-s − 4.96·43-s − 0.960·44-s − 9.01·46-s − 9.49·47-s + 15.7·49-s + 2.24·52-s + 6.84·53-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.80·7-s − 0.353·8-s − 0.289·11-s + 0.623·13-s − 1.27·14-s + 0.250·16-s + 0.0605·17-s + 0.229·19-s + 0.204·22-s + 1.88·23-s − 0.441·26-s + 0.901·28-s − 1.16·29-s + 0.531·31-s − 0.176·32-s − 0.0428·34-s − 0.00656·37-s − 0.162·38-s − 0.756·43-s − 0.144·44-s − 1.32·46-s − 1.38·47-s + 2.24·49-s + 0.311·52-s + 0.940·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.109835663\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.109835663\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 4.76T + 7T^{2} \) |
| 11 | \( 1 + 0.960T + 11T^{2} \) |
| 13 | \( 1 - 2.24T + 13T^{2} \) |
| 17 | \( 1 - 0.249T + 17T^{2} \) |
| 23 | \( 1 - 9.01T + 23T^{2} \) |
| 29 | \( 1 + 6.24T + 29T^{2} \) |
| 31 | \( 1 - 2.96T + 31T^{2} \) |
| 37 | \( 1 + 0.0399T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 4.96T + 43T^{2} \) |
| 47 | \( 1 + 9.49T + 47T^{2} \) |
| 53 | \( 1 - 6.84T + 53T^{2} \) |
| 59 | \( 1 - 14.5T + 59T^{2} \) |
| 61 | \( 1 - 7.53T + 61T^{2} \) |
| 67 | \( 1 - 5.72T + 67T^{2} \) |
| 71 | \( 1 - 9.61T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 6.07T + 79T^{2} \) |
| 83 | \( 1 + 7.45T + 83T^{2} \) |
| 89 | \( 1 + 4.07T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−7.932658215027898120725384839981, −7.21008328488719579784821187264, −6.68085154898125034539412318525, −5.48457584163991225089452602843, −5.21584799228143375362707506645, −4.31759631628925177117211266726, −3.39244050325859787080489787289, −2.39928089635169478746267491379, −1.58090406644084436406961904211, −0.859179324259370373552832318397,
0.859179324259370373552832318397, 1.58090406644084436406961904211, 2.39928089635169478746267491379, 3.39244050325859787080489787289, 4.31759631628925177117211266726, 5.21584799228143375362707506645, 5.48457584163991225089452602843, 6.68085154898125034539412318525, 7.21008328488719579784821187264, 7.932658215027898120725384839981