| L(s) = 1 | − 2.04·2-s + 3·3-s − 3.82·4-s − 12.0·5-s − 6.12·6-s − 29.7·7-s + 24.1·8-s + 9·9-s + 24.6·10-s − 28.0·11-s − 11.4·12-s + 60.7·14-s − 36.2·15-s − 18.7·16-s − 50.6·17-s − 18.3·18-s − 105.·19-s + 46.2·20-s − 89.2·21-s + 57.3·22-s − 160.·23-s + 72.4·24-s + 20.9·25-s + 27·27-s + 113.·28-s + 140.·29-s + 74.0·30-s + ⋯ |
| L(s) = 1 | − 0.722·2-s + 0.577·3-s − 0.478·4-s − 1.08·5-s − 0.416·6-s − 1.60·7-s + 1.06·8-s + 0.333·9-s + 0.780·10-s − 0.769·11-s − 0.276·12-s + 1.15·14-s − 0.623·15-s − 0.292·16-s − 0.722·17-s − 0.240·18-s − 1.26·19-s + 0.517·20-s − 0.927·21-s + 0.555·22-s − 1.45·23-s + 0.616·24-s + 0.167·25-s + 0.192·27-s + 0.768·28-s + 0.897·29-s + 0.450·30-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
| \(L(2)\) |
\(\approx\) |
\(0.2837534762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2837534762\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 2.04T + 8T^{2} \) |
| 5 | \( 1 + 12.0T + 125T^{2} \) |
| 7 | \( 1 + 29.7T + 343T^{2} \) |
| 11 | \( 1 + 28.0T + 1.33e3T^{2} \) |
| 17 | \( 1 + 50.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 105.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 160.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 140.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 228.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 295.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 192.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 36.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 438.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 286.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 537.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 102.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 75.5T + 3.89e5T^{2} \) |
| 79 | \( 1 - 17.5T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.46e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 334.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 748.T + 9.12e5T^{2} \) |
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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
−10.24781102541153325182718732055, −9.537176155319608814046380898982, −8.700209915987861492523847717149, −7.984635170979513700630639701958, −7.20195118050387779115268511079, −6.06315207404290797158305446601, −4.38195524046886459599132704104, −3.74783416555354025034339180164, −2.41995946035335524971548540966, −0.33755533381617029524801544648,
0.33755533381617029524801544648, 2.41995946035335524971548540966, 3.74783416555354025034339180164, 4.38195524046886459599132704104, 6.06315207404290797158305446601, 7.20195118050387779115268511079, 7.984635170979513700630639701958, 8.700209915987861492523847717149, 9.537176155319608814046380898982, 10.24781102541153325182718732055
