| L(s) = 1 | − i·2-s − 2i·3-s − 4-s − 2·6-s + i·8-s − 9-s + 3·11-s + 2i·12-s + 5i·13-s + 16-s − 6i·17-s + i·18-s + 19-s − 3i·22-s + 3i·23-s + 2·24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.15i·3-s − 0.5·4-s − 0.816·6-s + 0.353i·8-s − 0.333·9-s + 0.904·11-s + 0.577i·12-s + 1.38i·13-s + 0.250·16-s − 1.45i·17-s + 0.235i·18-s + 0.229·19-s − 0.639i·22-s + 0.625i·23-s + 0.408·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.725077825\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.725077825\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 5iT - 13T^{2} \) |
| 17 | \( 1 + 6iT - 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 3iT - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + 11iT - 37T^{2} \) |
| 41 | \( 1 - 3T + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 + 3iT - 47T^{2} \) |
| 53 | \( 1 - 3iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 + 12iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−8.912792581926207677104669623008, −7.67497637698218421622427193859, −7.11167948837869358277620588115, −6.51258983071990516320112485269, −5.48233414928129521784509526972, −4.45768563292731942658600692863, −3.65078736651990229680661268611, −2.42217099602311934520845320210, −1.69940574596552653217504142222, −0.67337244601238629872055702984,
1.21337527918625568825321105999, 3.01103752860562333010638456714, 3.81933650691979871125135077726, 4.54246182379169351962363756360, 5.29061019124048519003663550667, 6.16709069193190434481853565281, 6.78156099532944409876146794047, 8.043557958972856613391636881429, 8.331075190369435722032575783548, 9.353352723544831242184035705918
