L(s) = 1 | − 0.370i·2-s + i·3-s + 1.86·4-s + i·5-s + 0.370·6-s + (2.39 + 1.12i)7-s − 1.43i·8-s − 9-s + 0.370·10-s + (2.11 − 2.55i)11-s + 1.86i·12-s − 5.18·13-s + (0.415 − 0.888i)14-s − 15-s + 3.19·16-s + 2.13·17-s + ⋯ |
L(s) = 1 | − 0.262i·2-s + 0.577i·3-s + 0.931·4-s + 0.447i·5-s + 0.151·6-s + (0.905 + 0.424i)7-s − 0.506i·8-s − 0.333·9-s + 0.117·10-s + (0.636 − 0.771i)11-s + 0.537i·12-s − 1.43·13-s + (0.111 − 0.237i)14-s − 0.258·15-s + 0.798·16-s + 0.517·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 - 0.428i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.336172154\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.336172154\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-2.39 - 1.12i)T \) |
| 11 | \( 1 + (-2.11 + 2.55i)T \) |
good | 2 | \( 1 + 0.370iT - 2T^{2} \) |
| 13 | \( 1 + 5.18T + 13T^{2} \) |
| 17 | \( 1 - 2.13T + 17T^{2} \) |
| 19 | \( 1 - 5.11T + 19T^{2} \) |
| 23 | \( 1 - 7.91T + 23T^{2} \) |
| 29 | \( 1 + 7.81iT - 29T^{2} \) |
| 31 | \( 1 - 7.84iT - 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + 2.17T + 41T^{2} \) |
| 43 | \( 1 - 7.76iT - 43T^{2} \) |
| 47 | \( 1 - 5.62iT - 47T^{2} \) |
| 53 | \( 1 + 10.7T + 53T^{2} \) |
| 59 | \( 1 + 2.14iT - 59T^{2} \) |
| 61 | \( 1 + 8.26T + 61T^{2} \) |
| 67 | \( 1 + 2.80T + 67T^{2} \) |
| 71 | \( 1 - 9.32T + 71T^{2} \) |
| 73 | \( 1 - 10.0T + 73T^{2} \) |
| 79 | \( 1 - 8.98iT - 79T^{2} \) |
| 83 | \( 1 + 3.96T + 83T^{2} \) |
| 89 | \( 1 + 8.45iT - 89T^{2} \) |
| 97 | \( 1 - 8.31iT - 97T^{2} \) |
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
−9.888825967751309234798593796499, −9.262198277919302283648326430447, −8.135201728436926387943517036323, −7.39203674914537456116116777380, −6.55209771599573143757307275909, −5.50122886506470496623959059946, −4.78021521468329074180750549510, −3.33651131598566359887105989628, −2.74869624832430837502052006650, −1.39015792440269710044886346531,
1.22954754565115877821315439471, 2.11697320194248196958534128454, 3.37982765655300249654806666182, 4.93942857023753332051807394861, 5.31855186754072623369829217620, 6.73222005245062432818861982630, 7.32234077055353185060296420233, 7.70082442507467318011052963121, 8.811627990347595288447898588132, 9.703798685037530594215373973070