L(s) = 1 | − 2.49i·2-s + 1.66i·3-s − 4.22·4-s + (2.11 − 0.713i)5-s + 4.15·6-s + 4.24i·7-s + 5.56i·8-s + 0.229·9-s + (−1.78 − 5.28i)10-s − 6.39·11-s − 7.03i·12-s − 1.34i·13-s + 10.6·14-s + (1.18 + 3.52i)15-s + 5.42·16-s − 1.79i·17-s + ⋯ |
L(s) = 1 | − 1.76i·2-s + 0.960i·3-s − 2.11·4-s + (0.947 − 0.318i)5-s + 1.69·6-s + 1.60i·7-s + 1.96i·8-s + 0.0766·9-s + (−0.562 − 1.67i)10-s − 1.92·11-s − 2.03i·12-s − 0.371i·13-s + 2.83·14-s + (0.306 + 0.910i)15-s + 1.35·16-s − 0.436i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 - 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.231912585\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.231912585\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.11 + 0.713i)T \) |
| 241 | \( 1 - T \) |
good | 2 | \( 1 + 2.49iT - 2T^{2} \) |
| 3 | \( 1 - 1.66iT - 3T^{2} \) |
| 7 | \( 1 - 4.24iT - 7T^{2} \) |
| 11 | \( 1 + 6.39T + 11T^{2} \) |
| 13 | \( 1 + 1.34iT - 13T^{2} \) |
| 17 | \( 1 + 1.79iT - 17T^{2} \) |
| 19 | \( 1 - 3.80T + 19T^{2} \) |
| 23 | \( 1 - 5.33iT - 23T^{2} \) |
| 29 | \( 1 + 1.62T + 29T^{2} \) |
| 31 | \( 1 + 3.96T + 31T^{2} \) |
| 37 | \( 1 - 10.3iT - 37T^{2} \) |
| 41 | \( 1 - 2.25T + 41T^{2} \) |
| 43 | \( 1 - 1.94iT - 43T^{2} \) |
| 47 | \( 1 - 11.9iT - 47T^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 + 9.06T + 59T^{2} \) |
| 61 | \( 1 - 14.1T + 61T^{2} \) |
| 67 | \( 1 - 4.44iT - 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 + 4.44iT - 73T^{2} \) |
| 79 | \( 1 - 9.81T + 79T^{2} \) |
| 83 | \( 1 + 14.6iT - 83T^{2} \) |
| 89 | \( 1 + 10.4T + 89T^{2} \) |
| 97 | \( 1 + 5.74iT - 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.786008391241019418436018153887, −9.451416107846933364171858023191, −8.764022778402806504681556400693, −7.72810057764406647989966916026, −5.73898696652167977098847702189, −5.24297111771157448650296238684, −4.64855500116390292374632873489, −3.08459449956833720305817485474, −2.71684918335530251061743476076, −1.56582295555071590509591516011,
0.52745356082357524971387140857, 2.17210392769934774423813320699, 3.87087890687793302766815818448, 5.02655773264439605435817943753, 5.69080950557535166017059068193, 6.72686959813392096042058378817, 7.12089026604088273317211938590, 7.69525269842464895406951120886, 8.389972290646521264173056032593, 9.596122504113487707822065727659