| L(s) = 1 | − 2-s − 3.25·3-s + 4-s − 1.18·5-s + 3.25·6-s + 1.65·7-s − 8-s + 7.60·9-s + 1.18·10-s + 2.58·11-s − 3.25·12-s + 0.983·13-s − 1.65·14-s + 3.87·15-s + 16-s − 0.0738·17-s − 7.60·18-s − 2.86·19-s − 1.18·20-s − 5.40·21-s − 2.58·22-s + 4.30·23-s + 3.25·24-s − 3.58·25-s − 0.983·26-s − 15.0·27-s + 1.65·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.88·3-s + 0.5·4-s − 0.532·5-s + 1.32·6-s + 0.626·7-s − 0.353·8-s + 2.53·9-s + 0.376·10-s + 0.780·11-s − 0.940·12-s + 0.272·13-s − 0.443·14-s + 1.00·15-s + 0.250·16-s − 0.0179·17-s − 1.79·18-s − 0.657·19-s − 0.266·20-s − 1.17·21-s − 0.552·22-s + 0.898·23-s + 0.664·24-s − 0.716·25-s − 0.192·26-s − 2.88·27-s + 0.313·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 + 3.25T + 3T^{2} \) |
| 5 | \( 1 + 1.18T + 5T^{2} \) |
| 7 | \( 1 - 1.65T + 7T^{2} \) |
| 11 | \( 1 - 2.58T + 11T^{2} \) |
| 13 | \( 1 - 0.983T + 13T^{2} \) |
| 17 | \( 1 + 0.0738T + 17T^{2} \) |
| 19 | \( 1 + 2.86T + 19T^{2} \) |
| 23 | \( 1 - 4.30T + 23T^{2} \) |
| 29 | \( 1 + 2.02T + 29T^{2} \) |
| 31 | \( 1 + 8.35T + 31T^{2} \) |
| 37 | \( 1 + 0.973T + 37T^{2} \) |
| 41 | \( 1 + 7.12T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 + 2.03T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 6.74T + 59T^{2} \) |
| 61 | \( 1 - 7.19T + 61T^{2} \) |
| 67 | \( 1 + 1.26T + 67T^{2} \) |
| 71 | \( 1 - 12.6T + 71T^{2} \) |
| 73 | \( 1 - 2.45T + 73T^{2} \) |
| 79 | \( 1 + 4.99T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 2.78T + 89T^{2} \) |
| 97 | \( 1 + 5.58T + 97T^{2} \) |
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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
−7.948392488889862840498463489820, −7.16770620410750006768679955226, −6.72966001087532958373458622873, −5.86432043392367416814948522358, −5.29063156724995255331058405496, −4.36426153063030635506842307388, −3.70206308451073678416251072621, −1.92761366017962022007636953722, −1.07951322335179632217763898030, 0,
1.07951322335179632217763898030, 1.92761366017962022007636953722, 3.70206308451073678416251072621, 4.36426153063030635506842307388, 5.29063156724995255331058405496, 5.86432043392367416814948522358, 6.72966001087532958373458622873, 7.16770620410750006768679955226, 7.948392488889862840498463489820
