| L(s) = 1 | + 2.13·3-s − 1.07·5-s − 1.39·7-s + 1.54·9-s + 1.76·11-s + 2.85·13-s − 2.30·15-s − 1.86·17-s + 1.12·19-s − 2.97·21-s − 3.52·23-s − 3.83·25-s − 3.09·27-s + 7.79·29-s − 5.81·31-s + 3.77·33-s + 1.50·35-s − 3.03·37-s + 6.08·39-s − 10.2·41-s − 3.90·43-s − 1.67·45-s + 8.16·47-s − 5.05·49-s − 3.98·51-s − 14.3·53-s − 1.90·55-s + ⋯ |
| L(s) = 1 | + 1.23·3-s − 0.482·5-s − 0.527·7-s + 0.516·9-s + 0.533·11-s + 0.791·13-s − 0.594·15-s − 0.452·17-s + 0.256·19-s − 0.649·21-s − 0.734·23-s − 0.766·25-s − 0.595·27-s + 1.44·29-s − 1.04·31-s + 0.656·33-s + 0.254·35-s − 0.498·37-s + 0.975·39-s − 1.60·41-s − 0.594·43-s − 0.249·45-s + 1.19·47-s − 0.722·49-s − 0.557·51-s − 1.96·53-s − 0.257·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 2.13T + 3T^{2} \) |
| 5 | \( 1 + 1.07T + 5T^{2} \) |
| 7 | \( 1 + 1.39T + 7T^{2} \) |
| 11 | \( 1 - 1.76T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 + 1.86T + 17T^{2} \) |
| 19 | \( 1 - 1.12T + 19T^{2} \) |
| 23 | \( 1 + 3.52T + 23T^{2} \) |
| 29 | \( 1 - 7.79T + 29T^{2} \) |
| 31 | \( 1 + 5.81T + 31T^{2} \) |
| 37 | \( 1 + 3.03T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 3.90T + 43T^{2} \) |
| 47 | \( 1 - 8.16T + 47T^{2} \) |
| 53 | \( 1 + 14.3T + 53T^{2} \) |
| 59 | \( 1 + 2.80T + 59T^{2} \) |
| 61 | \( 1 + 1.45T + 61T^{2} \) |
| 67 | \( 1 + 4.93T + 67T^{2} \) |
| 71 | \( 1 - 14.8T + 71T^{2} \) |
| 73 | \( 1 - 13.2T + 73T^{2} \) |
| 79 | \( 1 - 9.94T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 + 13.1T + 89T^{2} \) |
| 97 | \( 1 - 1.76T + 97T^{2} \) |
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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
−7.75333109398565815183377760105, −6.75809881536288986886088813705, −6.35998410000863996916993956382, −5.37250832885756152101712592101, −4.39343567843109591612326660832, −3.57566966232634588715755064274, −3.36475667188090910623770612095, −2.30794747477598897711140483677, −1.47866989012179850486203970754, 0,
1.47866989012179850486203970754, 2.30794747477598897711140483677, 3.36475667188090910623770612095, 3.57566966232634588715755064274, 4.39343567843109591612326660832, 5.37250832885756152101712592101, 6.35998410000863996916993956382, 6.75809881536288986886088813705, 7.75333109398565815183377760105
