| L(s) = 1 | − 2.56·2-s − 0.178·3-s + 4.58·4-s + 0.458·6-s + 0.233·7-s − 6.64·8-s − 2.96·9-s − 2.70·11-s − 0.819·12-s + 4.02·13-s − 0.598·14-s + 7.88·16-s + 2.29·17-s + 7.61·18-s − 4.91·19-s − 0.0416·21-s + 6.94·22-s + 7.50·23-s + 1.18·24-s − 10.3·26-s + 1.06·27-s + 1.06·28-s + 4.49·29-s + 6.05·31-s − 6.95·32-s + 0.482·33-s − 5.89·34-s + ⋯ |
| L(s) = 1 | − 1.81·2-s − 0.103·3-s + 2.29·4-s + 0.187·6-s + 0.0881·7-s − 2.35·8-s − 0.989·9-s − 0.815·11-s − 0.236·12-s + 1.11·13-s − 0.159·14-s + 1.97·16-s + 0.556·17-s + 1.79·18-s − 1.12·19-s − 0.00908·21-s + 1.48·22-s + 1.56·23-s + 0.242·24-s − 2.02·26-s + 0.205·27-s + 0.202·28-s + 0.835·29-s + 1.08·31-s − 1.22·32-s + 0.0840·33-s − 1.01·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6463179278\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6463179278\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 + T \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 + 0.178T + 3T^{2} \) |
| 7 | \( 1 - 0.233T + 7T^{2} \) |
| 11 | \( 1 + 2.70T + 11T^{2} \) |
| 13 | \( 1 - 4.02T + 13T^{2} \) |
| 17 | \( 1 - 2.29T + 17T^{2} \) |
| 19 | \( 1 + 4.91T + 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 - 4.49T + 29T^{2} \) |
| 31 | \( 1 - 6.05T + 31T^{2} \) |
| 37 | \( 1 - 5.80T + 37T^{2} \) |
| 41 | \( 1 - 8.99T + 41T^{2} \) |
| 43 | \( 1 + 11.2T + 43T^{2} \) |
| 47 | \( 1 - 5.27T + 47T^{2} \) |
| 53 | \( 1 - 2.38T + 53T^{2} \) |
| 59 | \( 1 + 9.48T + 59T^{2} \) |
| 61 | \( 1 + 8.25T + 61T^{2} \) |
| 67 | \( 1 - 6.20T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 + 14.9T + 73T^{2} \) |
| 79 | \( 1 + 11.7T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 8.13T + 89T^{2} \) |
| 97 | \( 1 + 14.0T + 97T^{2} \) |
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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
−8.103979093715808163611494081869, −7.82522618104001845434127502764, −6.72891418541622574385297650130, −6.28611173261987375497744394306, −5.52774409816392151456053983053, −4.49264498008223931369644816279, −3.06145650712362434095611832543, −2.67111171301763835109498784173, −1.47237057240317157050271345374, −0.58479212297021854825642857276,
0.58479212297021854825642857276, 1.47237057240317157050271345374, 2.67111171301763835109498784173, 3.06145650712362434095611832543, 4.49264498008223931369644816279, 5.52774409816392151456053983053, 6.28611173261987375497744394306, 6.72891418541622574385297650130, 7.82522618104001845434127502764, 8.103979093715808163611494081869
