L(s) = 1 | + (−0.366 − 1.36i)2-s + 2i·3-s + (−1.73 + i)4-s + 3.46i·5-s + (2.73 − 0.732i)6-s − 1.26·7-s + (2 + 1.99i)8-s − 9-s + (4.73 − 1.26i)10-s − 4.73i·11-s + (−2 − 3.46i)12-s + i·13-s + (0.464 + 1.73i)14-s − 6.92·15-s + (1.99 − 3.46i)16-s + 5.46·17-s + ⋯ |
L(s) = 1 | + (−0.258 − 0.965i)2-s + 1.15i·3-s + (−0.866 + 0.5i)4-s + 1.54i·5-s + (1.11 − 0.298i)6-s − 0.479·7-s + (0.707 + 0.707i)8-s − 0.333·9-s + (1.49 − 0.400i)10-s − 1.42i·11-s + (−0.577 − 0.999i)12-s + 0.277i·13-s + (0.124 + 0.462i)14-s − 1.78·15-s + (0.499 − 0.866i)16-s + 1.32·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.756322 + 0.313279i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.756322 + 0.313279i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.366 + 1.36i)T \) |
| 13 | \( 1 - iT \) |
good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 - 3.46iT - 5T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 + 4.73iT - 11T^{2} \) |
| 17 | \( 1 - 5.46T + 17T^{2} \) |
| 19 | \( 1 - 0.732iT - 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2iT - 29T^{2} \) |
| 31 | \( 1 + 6.73T + 31T^{2} \) |
| 37 | \( 1 + 8.92iT - 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 + 0.535iT - 43T^{2} \) |
| 47 | \( 1 - 6.73T + 47T^{2} \) |
| 53 | \( 1 + 2.92iT - 53T^{2} \) |
| 59 | \( 1 + 10.1iT - 59T^{2} \) |
| 61 | \( 1 - 2.92iT - 61T^{2} \) |
| 67 | \( 1 - 0.732iT - 67T^{2} \) |
| 71 | \( 1 + 8.19T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 - 5.46T + 79T^{2} \) |
| 83 | \( 1 - 3.26iT - 83T^{2} \) |
| 89 | \( 1 + 17.3T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
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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
−14.13138761130749281911969637587, −12.74559147605110047710515783750, −11.22936593423537180748242896467, −10.80347344142736103279795786413, −9.927458358212439716120657493433, −8.999722199462404836567638118216, −7.39930033665639993270289510149, −5.65706916745960590564750799286, −3.75203968154316350580012571200, −3.06320050117788563455232785257,
1.23000020128735635730991750156, 4.56746430097494365243897227587, 5.76050998433423880724887136900, 7.15368383405723015950202175585, 7.88710400559734548488988068574, 9.093944912592433757917183976042, 9.975598714293025556706726442622, 12.22471881712449308840253616255, 12.77478134770588876714157957757, 13.41041458260834501861777971835