| L(s) = 1 | + 5.36·2-s + 3·3-s + 20.7·4-s − 2.69·5-s + 16.0·6-s − 15.2·7-s + 68.5·8-s + 9·9-s − 14.4·10-s + 66.8·11-s + 62.3·12-s − 81.5·14-s − 8.08·15-s + 201.·16-s + 4.16·17-s + 48.2·18-s − 26.0·19-s − 56.0·20-s − 45.6·21-s + 358.·22-s + 47.3·23-s + 205.·24-s − 117.·25-s + 27·27-s − 315.·28-s + 257.·29-s − 43.3·30-s + ⋯ |
| L(s) = 1 | + 1.89·2-s + 0.577·3-s + 2.59·4-s − 0.241·5-s + 1.09·6-s − 0.820·7-s + 3.03·8-s + 0.333·9-s − 0.457·10-s + 1.83·11-s + 1.49·12-s − 1.55·14-s − 0.139·15-s + 3.14·16-s + 0.0594·17-s + 0.632·18-s − 0.314·19-s − 0.626·20-s − 0.473·21-s + 3.47·22-s + 0.429·23-s + 1.74·24-s − 0.941·25-s + 0.192·27-s − 2.13·28-s + 1.64·29-s − 0.264·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(7.836215173\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.836215173\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 5.36T + 8T^{2} \) |
| 5 | \( 1 + 2.69T + 125T^{2} \) |
| 7 | \( 1 + 15.2T + 343T^{2} \) |
| 11 | \( 1 - 66.8T + 1.33e3T^{2} \) |
| 17 | \( 1 - 4.16T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.0T + 6.85e3T^{2} \) |
| 23 | \( 1 - 47.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 257.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 206.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 175.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 156.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 51.9T + 7.95e4T^{2} \) |
| 47 | \( 1 - 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 10.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 445.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 119.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 22.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 285.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 740.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 547.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 603.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 215.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.44e3T + 9.12e5T^{2} \) |
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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
−10.82943550905618116550137371191, −9.739377273958529009881797734727, −8.657024227064100914587953374746, −7.24424061570718059095758645112, −6.64103037606871112030999246610, −5.81766277428917450240952751626, −4.45478364512609268436053604670, −3.76205813531340362392371267046, −2.97351663310449526248611472019, −1.60735828916494866292647923818,
1.60735828916494866292647923818, 2.97351663310449526248611472019, 3.76205813531340362392371267046, 4.45478364512609268436053604670, 5.81766277428917450240952751626, 6.64103037606871112030999246610, 7.24424061570718059095758645112, 8.657024227064100914587953374746, 9.739377273958529009881797734727, 10.82943550905618116550137371191
