| L(s) = 1 | + 3.15·3-s + 4.23·5-s + 1.79·7-s + 6.95·9-s − 5.60·11-s + 6.12·13-s + 13.3·15-s − 3.98·17-s + 0.347·19-s + 5.67·21-s + 0.915·23-s + 12.9·25-s + 12.4·27-s + 3.33·29-s + 5.47·31-s − 17.6·33-s + 7.61·35-s + 0.150·37-s + 19.3·39-s − 7.39·41-s − 6.66·43-s + 29.4·45-s + 4.76·47-s − 3.76·49-s − 12.5·51-s − 5.03·53-s − 23.7·55-s + ⋯ |
| L(s) = 1 | + 1.82·3-s + 1.89·5-s + 0.679·7-s + 2.31·9-s − 1.68·11-s + 1.69·13-s + 3.44·15-s − 0.966·17-s + 0.0796·19-s + 1.23·21-s + 0.190·23-s + 2.58·25-s + 2.39·27-s + 0.618·29-s + 0.982·31-s − 3.07·33-s + 1.28·35-s + 0.0247·37-s + 3.09·39-s − 1.15·41-s − 1.01·43-s + 4.38·45-s + 0.695·47-s − 0.537·49-s − 1.76·51-s − 0.691·53-s − 3.19·55-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
| \(L(1)\) |
\(\approx\) |
\(6.823186211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.823186211\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.15T + 3T^{2} \) |
| 5 | \( 1 - 4.23T + 5T^{2} \) |
| 7 | \( 1 - 1.79T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 + 3.98T + 17T^{2} \) |
| 19 | \( 1 - 0.347T + 19T^{2} \) |
| 23 | \( 1 - 0.915T + 23T^{2} \) |
| 29 | \( 1 - 3.33T + 29T^{2} \) |
| 31 | \( 1 - 5.47T + 31T^{2} \) |
| 37 | \( 1 - 0.150T + 37T^{2} \) |
| 41 | \( 1 + 7.39T + 41T^{2} \) |
| 43 | \( 1 + 6.66T + 43T^{2} \) |
| 47 | \( 1 - 4.76T + 47T^{2} \) |
| 53 | \( 1 + 5.03T + 53T^{2} \) |
| 59 | \( 1 + 3.74T + 59T^{2} \) |
| 61 | \( 1 - 8.31T + 61T^{2} \) |
| 67 | \( 1 + 7.51T + 67T^{2} \) |
| 71 | \( 1 + 15.1T + 71T^{2} \) |
| 73 | \( 1 + 12.8T + 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 + 17.6T + 83T^{2} \) |
| 89 | \( 1 + 4.69T + 89T^{2} \) |
| 97 | \( 1 - 4.06T + 97T^{2} \) |
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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
−8.155511431442425602740004156964, −7.23877127052189731896671845342, −6.50184129418125938633403068506, −5.77071307765258509936988578809, −4.95025619198389067817625817209, −4.30213215567119312445954228774, −2.98423233715190286684114245780, −2.79694951918871993764903624657, −1.81264815258404985951696011707, −1.42298042129291639795112181588,
1.42298042129291639795112181588, 1.81264815258404985951696011707, 2.79694951918871993764903624657, 2.98423233715190286684114245780, 4.30213215567119312445954228774, 4.95025619198389067817625817209, 5.77071307765258509936988578809, 6.50184129418125938633403068506, 7.23877127052189731896671845342, 8.155511431442425602740004156964
