| L(s) = 1 | + 2·2-s − 4.45·3-s + 4·4-s + 11.8·5-s − 8.91·6-s + 8·8-s − 7.11·9-s + 23.7·10-s − 10.2·11-s − 17.8·12-s − 43.4·13-s − 53.0·15-s + 16·16-s + 54.9·17-s − 14.2·18-s − 47.0·19-s + 47.5·20-s − 20.5·22-s − 23·23-s − 35.6·24-s + 16.2·25-s − 86.8·26-s + 152.·27-s + 245.·29-s − 106.·30-s − 287.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.858·3-s + 0.5·4-s + 1.06·5-s − 0.606·6-s + 0.353·8-s − 0.263·9-s + 0.751·10-s − 0.281·11-s − 0.429·12-s − 0.926·13-s − 0.912·15-s + 0.250·16-s + 0.783·17-s − 0.186·18-s − 0.568·19-s + 0.531·20-s − 0.199·22-s − 0.208·23-s − 0.303·24-s + 0.130·25-s − 0.655·26-s + 1.08·27-s + 1.57·29-s − 0.645·30-s − 1.66·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2254 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.687966032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.687966032\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 4.45T + 27T^{2} \) |
| 5 | \( 1 - 11.8T + 125T^{2} \) |
| 11 | \( 1 + 10.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 43.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 54.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 47.0T + 6.85e3T^{2} \) |
| 29 | \( 1 - 245.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 287.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 35.0T + 6.89e4T^{2} \) |
| 43 | \( 1 - 257.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 618.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 349.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 626.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 626.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 435.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 727.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 460.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 322.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 995.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.16e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 549.T + 9.12e5T^{2} \) |
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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
−8.728416715724896787267176274598, −7.68951111535851928062383269888, −6.82537744419262466086450128416, −6.12418720313706907276671635050, −5.40633388186350022835148430331, −5.08611693760333775693247204751, −3.91850517280330753103722650715, −2.73143894027716609531787670581, −2.00122966879933824463611880601, −0.67206942106985435977884088917,
0.67206942106985435977884088917, 2.00122966879933824463611880601, 2.73143894027716609531787670581, 3.91850517280330753103722650715, 5.08611693760333775693247204751, 5.40633388186350022835148430331, 6.12418720313706907276671635050, 6.82537744419262466086450128416, 7.68951111535851928062383269888, 8.728416715724896787267176274598
