| L(s) = 1 | + 1.73i·3-s + (4.90 − 4.99i)7-s − 2.99·9-s + 3.52·11-s + 1.16i·13-s − 3.02i·17-s − 16.3i·19-s + (8.64 + 8.50i)21-s + 16.0·23-s − 5.19i·27-s + 0.746·29-s + 29.1i·31-s + 6.10i·33-s − 29.5·37-s − 2.01·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (0.701 − 0.712i)7-s − 0.333·9-s + 0.320·11-s + 0.0896i·13-s − 0.177i·17-s − 0.858i·19-s + (0.411 + 0.404i)21-s + 0.697·23-s − 0.192i·27-s + 0.0257·29-s + 0.938i·31-s + 0.185i·33-s − 0.798·37-s − 0.0517·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.712 + 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.011008866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.011008866\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 1.73iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-4.90 + 4.99i)T \) |
| good | 11 | \( 1 - 3.52T + 121T^{2} \) |
| 13 | \( 1 - 1.16iT - 169T^{2} \) |
| 17 | \( 1 + 3.02iT - 289T^{2} \) |
| 19 | \( 1 + 16.3iT - 361T^{2} \) |
| 23 | \( 1 - 16.0T + 529T^{2} \) |
| 29 | \( 1 - 0.746T + 841T^{2} \) |
| 31 | \( 1 - 29.1iT - 961T^{2} \) |
| 37 | \( 1 + 29.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 39.0iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 27.0T + 1.84e3T^{2} \) |
| 47 | \( 1 + 45.4iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 33.1T + 2.80e3T^{2} \) |
| 59 | \( 1 + 60.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 29.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 68.8T + 4.48e3T^{2} \) |
| 71 | \( 1 - 79.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 62.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 114.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 1.64iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 36.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 84.0iT - 9.40e3T^{2} \) |
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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
−8.907521999054178196583688083578, −8.141095935876025651511529563407, −7.20586270578818528780123127317, −6.63470664785132600102066997297, −5.38008222321933832749468033024, −4.80417492200766642503301906178, −3.97053045026559408181147114148, −3.08542634608459486342290959008, −1.80715589142943967350152592612, −0.55646078375958370122948858155,
1.10456635619813145106391111789, 2.02681900891427042301002452853, 3.01305180369874653584502451996, 4.15934283112644240122330321078, 5.14174776244515443925491668490, 5.90378793744123290205033248820, 6.60981839793412261796251392145, 7.65267551504313598673238138707, 8.116909936378158776433014500064, 8.954550948824076556162184341697
