| L(s) = 1 | + 4.53·2-s − 3.57·3-s + 12.5·4-s + 13.4·5-s − 16.2·6-s + 20.5·8-s − 14.1·9-s + 61.0·10-s + 0.813·11-s − 44.8·12-s − 34.9·13-s − 48.2·15-s − 7.21·16-s − 117.·17-s − 64.2·18-s + 93.2·19-s + 168.·20-s + 3.68·22-s + 120.·23-s − 73.4·24-s + 56.6·25-s − 158.·26-s + 147.·27-s + 8.56·29-s − 218.·30-s + 82.1·31-s − 196.·32-s + ⋯ |
| L(s) = 1 | + 1.60·2-s − 0.688·3-s + 1.56·4-s + 1.20·5-s − 1.10·6-s + 0.907·8-s − 0.525·9-s + 1.93·10-s + 0.0222·11-s − 1.07·12-s − 0.745·13-s − 0.830·15-s − 0.112·16-s − 1.67·17-s − 0.841·18-s + 1.12·19-s + 1.88·20-s + 0.0357·22-s + 1.09·23-s − 0.625·24-s + 0.453·25-s − 1.19·26-s + 1.05·27-s + 0.0548·29-s − 1.33·30-s + 0.475·31-s − 1.08·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.642588989\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.642588989\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 - 4.53T + 8T^{2} \) |
| 3 | \( 1 + 3.57T + 27T^{2} \) |
| 5 | \( 1 - 13.4T + 125T^{2} \) |
| 11 | \( 1 - 0.813T + 1.33e3T^{2} \) |
| 13 | \( 1 + 34.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 93.2T + 6.85e3T^{2} \) |
| 23 | \( 1 - 120.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 8.56T + 2.43e4T^{2} \) |
| 31 | \( 1 - 82.1T + 2.97e4T^{2} \) |
| 37 | \( 1 - 28.8T + 5.06e4T^{2} \) |
| 41 | \( 1 - 70.5T + 6.89e4T^{2} \) |
| 43 | \( 1 - 417.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 338.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 94.1T + 2.05e5T^{2} \) |
| 61 | \( 1 - 120.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 792.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 449.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 469.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.01e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 104.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.57e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 550.T + 9.12e5T^{2} \) |
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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
−14.76062167306981195962056591561, −13.82133345908655903818731112287, −13.04231668804884203859304812969, −11.86280842489651821316430434470, −10.88046747794532624205136787868, −9.273059859876872027562328900671, −6.78879952285735819583181769122, −5.75166282243998565089783927738, −4.80634343790062199057737752864, −2.61622532863461852539181307197,
2.61622532863461852539181307197, 4.80634343790062199057737752864, 5.75166282243998565089783927738, 6.78879952285735819583181769122, 9.273059859876872027562328900671, 10.88046747794532624205136787868, 11.86280842489651821316430434470, 13.04231668804884203859304812969, 13.82133345908655903818731112287, 14.76062167306981195962056591561
