| L(s) = 1 | − 3.31·3-s + 16.0·5-s − 35.2·7-s − 16.0·9-s − 35.8·11-s − 26.4·13-s − 53.2·15-s − 92.3·17-s − 71.9·19-s + 116.·21-s − 23·23-s + 133.·25-s + 142.·27-s − 131.·29-s − 330.·31-s + 118.·33-s − 567.·35-s + 55.6·37-s + 87.7·39-s + 403.·41-s + 289.·43-s − 257.·45-s − 141.·47-s + 902.·49-s + 305.·51-s − 0.542·53-s − 575.·55-s + ⋯ |
| L(s) = 1 | − 0.637·3-s + 1.43·5-s − 1.90·7-s − 0.593·9-s − 0.981·11-s − 0.564·13-s − 0.916·15-s − 1.31·17-s − 0.868·19-s + 1.21·21-s − 0.208·23-s + 1.06·25-s + 1.01·27-s − 0.843·29-s − 1.91·31-s + 0.626·33-s − 2.73·35-s + 0.247·37-s + 0.360·39-s + 1.53·41-s + 1.02·43-s − 0.852·45-s − 0.437·47-s + 2.63·49-s + 0.840·51-s − 0.00140·53-s − 1.41·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.4579376367\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4579376367\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 3.31T + 27T^{2} \) |
| 5 | \( 1 - 16.0T + 125T^{2} \) |
| 7 | \( 1 + 35.2T + 343T^{2} \) |
| 11 | \( 1 + 35.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + 92.3T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.9T + 6.85e3T^{2} \) |
| 29 | \( 1 + 131.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 330.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 55.6T + 5.06e4T^{2} \) |
| 41 | \( 1 - 403.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 289.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 141.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 0.542T + 1.48e5T^{2} \) |
| 59 | \( 1 - 241.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 779.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 324.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 408.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 124.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 896.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 508.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 908.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 549.T + 9.12e5T^{2} \) |
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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
−9.414887058418915723685975293661, −8.598215421177994168386405420923, −7.17025295705897801157518569542, −6.53220585721036871525081425905, −5.76819330530701842280811486776, −5.44490705093802668151226381259, −4.04961379771621764938933460057, −2.71521348117260931629065636392, −2.23166037224200050137764498400, −0.31263595085956981523517205110,
0.31263595085956981523517205110, 2.23166037224200050137764498400, 2.71521348117260931629065636392, 4.04961379771621764938933460057, 5.44490705093802668151226381259, 5.76819330530701842280811486776, 6.53220585721036871525081425905, 7.17025295705897801157518569542, 8.598215421177994168386405420923, 9.414887058418915723685975293661
