| L(s) = 1 | + 18.1·5-s − 0.822·7-s − 65.5·11-s + 31.8·13-s + 124.·17-s + 27.8·19-s − 43.1·23-s + 205.·25-s + 39.8·29-s + 294.·31-s − 14.9·35-s − 104.·37-s + 307.·41-s − 361.·43-s + 397.·47-s − 342.·49-s + 107.·53-s − 1.19e3·55-s + 188.·59-s − 99.0·61-s + 578.·65-s + 425.·67-s − 445.·71-s − 499.·73-s + 53.9·77-s + 570.·79-s + 1.30e3·83-s + ⋯ |
| L(s) = 1 | + 1.62·5-s − 0.0444·7-s − 1.79·11-s + 0.678·13-s + 1.77·17-s + 0.336·19-s − 0.391·23-s + 1.64·25-s + 0.254·29-s + 1.70·31-s − 0.0722·35-s − 0.462·37-s + 1.17·41-s − 1.28·43-s + 1.23·47-s − 0.998·49-s + 0.277·53-s − 2.92·55-s + 0.416·59-s − 0.207·61-s + 1.10·65-s + 0.775·67-s − 0.744·71-s − 0.801·73-s + 0.0797·77-s + 0.812·79-s + 1.73·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.759846921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.759846921\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 18.1T + 125T^{2} \) |
| 7 | \( 1 + 0.822T + 343T^{2} \) |
| 11 | \( 1 + 65.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 31.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 124.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 27.8T + 6.85e3T^{2} \) |
| 23 | \( 1 + 43.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 39.8T + 2.43e4T^{2} \) |
| 31 | \( 1 - 294.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 104.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 307.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 361.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 397.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 107.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 188.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 99.0T + 2.26e5T^{2} \) |
| 67 | \( 1 - 425.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 445.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 499.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 570.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.32e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 944.T + 9.12e5T^{2} \) |
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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
−10.19664236217597114654959457423, −9.538503742657354749242245282953, −8.380322030442082804821663659452, −7.62761880394120755188408956815, −6.32012786656726595919978322754, −5.62965643826621142352220091362, −4.96063606658122677943803049197, −3.21727699074220159934240217241, −2.30064946173452394325257242570, −1.02290660367226557313468749690,
1.02290660367226557313468749690, 2.30064946173452394325257242570, 3.21727699074220159934240217241, 4.96063606658122677943803049197, 5.62965643826621142352220091362, 6.32012786656726595919978322754, 7.62761880394120755188408956815, 8.380322030442082804821663659452, 9.538503742657354749242245282953, 10.19664236217597114654959457423
