| L(s) = 1 | + 0.594·3-s + 2.93·5-s − 0.580·7-s − 2.64·9-s + 11-s + 3.41·13-s + 1.74·15-s − 4.86·17-s − 7.84·19-s − 0.345·21-s − 6.74·23-s + 3.59·25-s − 3.35·27-s + 3.88·29-s + 1.19·31-s + 0.594·33-s − 1.70·35-s + 9.57·37-s + 2.03·39-s − 3.80·41-s − 0.346·43-s − 7.75·45-s − 9.28·47-s − 6.66·49-s − 2.89·51-s − 10.2·53-s + 2.93·55-s + ⋯ |
| L(s) = 1 | + 0.343·3-s + 1.31·5-s − 0.219·7-s − 0.882·9-s + 0.301·11-s + 0.946·13-s + 0.450·15-s − 1.17·17-s − 1.79·19-s − 0.0753·21-s − 1.40·23-s + 0.719·25-s − 0.646·27-s + 0.720·29-s + 0.214·31-s + 0.103·33-s − 0.287·35-s + 1.57·37-s + 0.325·39-s − 0.594·41-s − 0.0527·43-s − 1.15·45-s − 1.35·47-s − 0.951·49-s − 0.404·51-s − 1.40·53-s + 0.395·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6028 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 - T \) |
| 137 | \( 1 + T \) |
| good | 3 | \( 1 - 0.594T + 3T^{2} \) |
| 5 | \( 1 - 2.93T + 5T^{2} \) |
| 7 | \( 1 + 0.580T + 7T^{2} \) |
| 13 | \( 1 - 3.41T + 13T^{2} \) |
| 17 | \( 1 + 4.86T + 17T^{2} \) |
| 19 | \( 1 + 7.84T + 19T^{2} \) |
| 23 | \( 1 + 6.74T + 23T^{2} \) |
| 29 | \( 1 - 3.88T + 29T^{2} \) |
| 31 | \( 1 - 1.19T + 31T^{2} \) |
| 37 | \( 1 - 9.57T + 37T^{2} \) |
| 41 | \( 1 + 3.80T + 41T^{2} \) |
| 43 | \( 1 + 0.346T + 43T^{2} \) |
| 47 | \( 1 + 9.28T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 9.61T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 - 6.03T + 67T^{2} \) |
| 71 | \( 1 + 8.44T + 71T^{2} \) |
| 73 | \( 1 - 3.59T + 73T^{2} \) |
| 79 | \( 1 + 8.89T + 79T^{2} \) |
| 83 | \( 1 + 5.35T + 83T^{2} \) |
| 89 | \( 1 - 6.75T + 89T^{2} \) |
| 97 | \( 1 + 0.768T + 97T^{2} \) |
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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
−8.064577557955855128728367995751, −6.62323777830571984088396152721, −6.25253213617016273461387547876, −5.92884903287941264391670857363, −4.78131610282872209170065758538, −4.07187028159253964847863395351, −3.05461947056493503029077500556, −2.24702097353742518036845606724, −1.63883402614857273506790946797, 0,
1.63883402614857273506790946797, 2.24702097353742518036845606724, 3.05461947056493503029077500556, 4.07187028159253964847863395351, 4.78131610282872209170065758538, 5.92884903287941264391670857363, 6.25253213617016273461387547876, 6.62323777830571984088396152721, 8.064577557955855128728367995751
