| L(s) = 1 | − 3-s − 1.41·5-s − 4.24·7-s + 9-s + 6·11-s − 5.65·13-s + 1.41·15-s − 6·17-s + 4·19-s + 4.24·21-s + 2.82·23-s − 2.99·25-s − 27-s − 1.41·29-s − 1.41·31-s − 6·33-s + 6·35-s + 8.48·37-s + 5.65·39-s − 2·41-s − 1.41·45-s + 2.82·47-s + 10.9·49-s + 6·51-s + 9.89·53-s − 8.48·55-s − 4·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.632·5-s − 1.60·7-s + 0.333·9-s + 1.80·11-s − 1.56·13-s + 0.365·15-s − 1.45·17-s + 0.917·19-s + 0.925·21-s + 0.589·23-s − 0.599·25-s − 0.192·27-s − 0.262·29-s − 0.254·31-s − 1.04·33-s + 1.01·35-s + 1.39·37-s + 0.905·39-s − 0.312·41-s − 0.210·45-s + 0.412·47-s + 1.57·49-s + 0.840·51-s + 1.35·53-s − 1.14·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7798231359\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7798231359\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 + 1.41T + 5T^{2} \) |
| 7 | \( 1 + 4.24T + 7T^{2} \) |
| 11 | \( 1 - 6T + 11T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 1.41T + 31T^{2} \) |
| 37 | \( 1 - 8.48T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 9.89T + 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 + 2.82T + 71T^{2} \) |
| 73 | \( 1 - 8T + 73T^{2} \) |
| 79 | \( 1 - 12.7T + 79T^{2} \) |
| 83 | \( 1 - 2T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−9.479147082819314286887374233082, −8.974363245026290771882010558512, −7.58929421185305713271870038264, −6.82330659221935284042956344159, −6.49327045770963959786689606839, −5.36855720358405827873265334204, −4.25361943037059161249668463097, −3.63660714636738948326101316675, −2.40912332359023244618515680324, −0.61971493429863042851907788793,
0.61971493429863042851907788793, 2.40912332359023244618515680324, 3.63660714636738948326101316675, 4.25361943037059161249668463097, 5.36855720358405827873265334204, 6.49327045770963959786689606839, 6.82330659221935284042956344159, 7.58929421185305713271870038264, 8.974363245026290771882010558512, 9.479147082819314286887374233082
