| L(s) = 1 | + 2.42·3-s + 3.69·5-s − 4.84·7-s + 2.85·9-s − 11-s − 3.71·13-s + 8.93·15-s − 3.07·17-s − 2.32·19-s − 11.7·21-s − 3.41·23-s + 8.63·25-s − 0.343·27-s + 2.45·29-s − 8.99·31-s − 2.42·33-s − 17.8·35-s + 5.73·37-s − 8.99·39-s − 1.72·41-s − 8.48·43-s + 10.5·45-s − 3.91·47-s + 16.4·49-s − 7.45·51-s + 9.81·53-s − 3.69·55-s + ⋯ |
| L(s) = 1 | + 1.39·3-s + 1.65·5-s − 1.83·7-s + 0.952·9-s − 0.301·11-s − 1.03·13-s + 2.30·15-s − 0.746·17-s − 0.533·19-s − 2.55·21-s − 0.712·23-s + 1.72·25-s − 0.0661·27-s + 0.456·29-s − 1.61·31-s − 0.421·33-s − 3.02·35-s + 0.943·37-s − 1.43·39-s − 0.269·41-s − 1.29·43-s + 1.57·45-s − 0.571·47-s + 2.34·49-s − 1.04·51-s + 1.34·53-s − 0.497·55-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 | \( 1 - 2.42T + 3T^{2} \) |
| 5 | \( 1 - 3.69T + 5T^{2} \) |
| 7 | \( 1 + 4.84T + 7T^{2} \) |
| 13 | \( 1 + 3.71T + 13T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 19 | \( 1 + 2.32T + 19T^{2} \) |
| 23 | \( 1 + 3.41T + 23T^{2} \) |
| 29 | \( 1 - 2.45T + 29T^{2} \) |
| 31 | \( 1 + 8.99T + 31T^{2} \) |
| 37 | \( 1 - 5.73T + 37T^{2} \) |
| 41 | \( 1 + 1.72T + 41T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + 3.91T + 47T^{2} \) |
| 53 | \( 1 - 9.81T + 53T^{2} \) |
| 59 | \( 1 + 5.78T + 59T^{2} \) |
| 61 | \( 1 + 5.85T + 61T^{2} \) |
| 67 | \( 1 + 11.7T + 67T^{2} \) |
| 71 | \( 1 + 7.66T + 71T^{2} \) |
| 73 | \( 1 + 0.840T + 73T^{2} \) |
| 79 | \( 1 - 12.1T + 79T^{2} \) |
| 83 | \( 1 - 10.0T + 83T^{2} \) |
| 89 | \( 1 - 1.04T + 89T^{2} \) |
| 97 | \( 1 - 3.31T + 97T^{2} \) |
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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
−7.71602806744179107923830191159, −6.97444673804088124050863529677, −6.33378547090490848347186786531, −5.78993519656333119895597430974, −4.80826659410546633275560528498, −3.75332277915347052713009824475, −2.97409828930402410198829601488, −2.42105408396960370390029427598, −1.84339738249254240108475014973, 0,
1.84339738249254240108475014973, 2.42105408396960370390029427598, 2.97409828930402410198829601488, 3.75332277915347052713009824475, 4.80826659410546633275560528498, 5.78993519656333119895597430974, 6.33378547090490848347186786531, 6.97444673804088124050863529677, 7.71602806744179107923830191159
