L(s) = 1 | + 2.22·3-s − 1.09·5-s + 1.95·7-s + 1.94·9-s + 2.63·11-s − 0.307·13-s − 2.43·15-s − 5.78·17-s − 0.355·19-s + 4.34·21-s + 1.82·23-s − 3.79·25-s − 2.35·27-s − 9.46·29-s − 8.77·31-s + 5.84·33-s − 2.14·35-s + 0.369·37-s − 0.684·39-s − 5.32·41-s − 4.57·43-s − 2.12·45-s + 4.16·47-s − 3.17·49-s − 12.8·51-s + 3.30·53-s − 2.88·55-s + ⋯ |
L(s) = 1 | + 1.28·3-s − 0.490·5-s + 0.739·7-s + 0.647·9-s + 0.793·11-s − 0.0853·13-s − 0.629·15-s − 1.40·17-s − 0.0815·19-s + 0.948·21-s + 0.379·23-s − 0.759·25-s − 0.452·27-s − 1.75·29-s − 1.57·31-s + 1.01·33-s − 0.362·35-s + 0.0608·37-s − 0.109·39-s − 0.832·41-s − 0.697·43-s − 0.317·45-s + 0.606·47-s − 0.453·49-s − 1.79·51-s + 0.454·53-s − 0.388·55-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 | \( 1 - 2.22T + 3T^{2} \) |
| 5 | \( 1 + 1.09T + 5T^{2} \) |
| 7 | \( 1 - 1.95T + 7T^{2} \) |
| 11 | \( 1 - 2.63T + 11T^{2} \) |
| 13 | \( 1 + 0.307T + 13T^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 + 0.355T + 19T^{2} \) |
| 23 | \( 1 - 1.82T + 23T^{2} \) |
| 29 | \( 1 + 9.46T + 29T^{2} \) |
| 31 | \( 1 + 8.77T + 31T^{2} \) |
| 37 | \( 1 - 0.369T + 37T^{2} \) |
| 41 | \( 1 + 5.32T + 41T^{2} \) |
| 43 | \( 1 + 4.57T + 43T^{2} \) |
| 47 | \( 1 - 4.16T + 47T^{2} \) |
| 53 | \( 1 - 3.30T + 53T^{2} \) |
| 59 | \( 1 - 7.66T + 59T^{2} \) |
| 61 | \( 1 + 8.11T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 9.39T + 71T^{2} \) |
| 73 | \( 1 - 6.49T + 73T^{2} \) |
| 79 | \( 1 - 5.39T + 79T^{2} \) |
| 83 | \( 1 + 13.8T + 83T^{2} \) |
| 89 | \( 1 - 3.57T + 89T^{2} \) |
| 97 | \( 1 - 6.26T + 97T^{2} \) |
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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
−7.45411965874059342820217471374, −7.19471637472731921833749535706, −6.16494022870483255105804649002, −5.30233103611344369007225191085, −4.37759274750289868281445447352, −3.84779639684692345108125160087, −3.20748847467656406795661476718, −2.08458979182525022920408211935, −1.69724682688250925108180004380, 0,
1.69724682688250925108180004380, 2.08458979182525022920408211935, 3.20748847467656406795661476718, 3.84779639684692345108125160087, 4.37759274750289868281445447352, 5.30233103611344369007225191085, 6.16494022870483255105804649002, 7.19471637472731921833749535706, 7.45411965874059342820217471374