L(s) = 1 | − i·5-s − 2·11-s − 25-s − 2i·29-s − 2i·31-s − 49-s + 2i·55-s − 2·59-s + 2i·79-s − 2i·101-s + ⋯ |
L(s) = 1 | − i·5-s − 2·11-s − 25-s − 2i·29-s − 2i·31-s − 49-s + 2i·55-s − 2·59-s + 2i·79-s − 2i·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6699649747\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6699649747\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + 2T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + 2iT - T^{2} \) |
| 31 | \( 1 + 2iT - T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - 2iT - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.475054487925109906245152847210, −7.949759081703459017712212768714, −7.50341078341659257465136480910, −6.13492548583102592963685286850, −5.59410818043879712713621234199, −4.77873223941865013788183837483, −4.11531973979706627298228323569, −2.82405056728476985418503660155, −1.99146617552321603127566000554, −0.38898812579356765423182752513,
1.78842535586474670052568472846, 2.94774358625620408474538972010, 3.29257989099249375954507285473, 4.76319198064855138069403366956, 5.31811541550600676038158744916, 6.26646123443303154053973883447, 7.07518669611979680886792163217, 7.64280772688851557876122890586, 8.375646719949536093022274072792, 9.228213744599735915277949965899