L(s) = 1 | − i·2-s + (0.707 − 0.707i)3-s − 4-s + (1.88 − 1.19i)5-s + (−0.707 − 0.707i)6-s + (−1.50 + 1.50i)7-s + i·8-s − 1.00i·9-s + (−1.19 − 1.88i)10-s − 0.249i·11-s + (−0.707 + 0.707i)12-s − 2.83i·13-s + (1.50 + 1.50i)14-s + (0.486 − 2.18i)15-s + 16-s − 7.29·17-s + ⋯ |
L(s) = 1 | − 0.707i·2-s + (0.408 − 0.408i)3-s − 0.5·4-s + (0.843 − 0.536i)5-s + (−0.288 − 0.288i)6-s + (−0.569 + 0.569i)7-s + 0.353i·8-s − 0.333i·9-s + (−0.379 − 0.596i)10-s − 0.0750i·11-s + (−0.204 + 0.204i)12-s − 0.785i·13-s + (0.402 + 0.402i)14-s + (0.125 − 0.563i)15-s + 0.250·16-s − 1.76·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.897 + 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.550397382\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.550397382\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 + (-1.88 + 1.19i)T \) |
| 37 | \( 1 + (-3.06 - 5.25i)T \) |
good | 7 | \( 1 + (1.50 - 1.50i)T - 7iT^{2} \) |
| 11 | \( 1 + 0.249iT - 11T^{2} \) |
| 13 | \( 1 + 2.83iT - 13T^{2} \) |
| 17 | \( 1 + 7.29T + 17T^{2} \) |
| 19 | \( 1 + (-1.89 + 1.89i)T - 19iT^{2} \) |
| 23 | \( 1 + 8.28iT - 23T^{2} \) |
| 29 | \( 1 + (2.37 + 2.37i)T + 29iT^{2} \) |
| 31 | \( 1 + (-1.83 + 1.83i)T - 31iT^{2} \) |
| 41 | \( 1 + 10.3iT - 41T^{2} \) |
| 43 | \( 1 + 7.61iT - 43T^{2} \) |
| 47 | \( 1 + (2.82 - 2.82i)T - 47iT^{2} \) |
| 53 | \( 1 + (-4.20 - 4.20i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.26 + 6.26i)T - 59iT^{2} \) |
| 61 | \( 1 + (3.22 - 3.22i)T - 61iT^{2} \) |
| 67 | \( 1 + (-5.68 - 5.68i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.37T + 71T^{2} \) |
| 73 | \( 1 + (7.48 - 7.48i)T - 73iT^{2} \) |
| 79 | \( 1 + (-5.99 + 5.99i)T - 79iT^{2} \) |
| 83 | \( 1 + (-5.98 - 5.98i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.37 - 5.37i)T + 89iT^{2} \) |
| 97 | \( 1 + 5.94T + 97T^{2} \) |
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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
−9.393895895503877232864329607770, −8.840782687954443874993244259962, −8.226004090104069286540937253946, −6.85656074171623141410605138276, −6.08976899713966693314969806823, −5.13649681451480286314736042718, −4.11159907246212023731196184794, −2.68969939861315460337334592354, −2.23003313185781589363529135746, −0.62857953605653630307369516030,
1.81577798603589130579050024717, 3.16263771028969537300470129871, 4.10434135249204026678969533656, 5.11200410598659644954597762957, 6.16358219010872699289292996862, 6.82871718619604034481863009576, 7.53279937158269740402100537461, 8.668448309773174873290898755488, 9.549350640011765648483255343176, 9.747419987605854055803152998025