L(s) = 1 | + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (0.896 − 0.142i)17-s + (1.11 + 1.53i)29-s + (0.809 + 1.58i)37-s + (−0.363 + 1.11i)41-s + i·49-s + (−0.309 − 0.0489i)53-s + (0.363 + 1.11i)61-s + (−0.142 + 0.278i)73-s + (0.809 + 0.587i)81-s + (−1.80 + 0.587i)89-s + (0.278 − 1.76i)97-s − 0.618·101-s + (−1.53 − 0.5i)109-s + ⋯ |
L(s) = 1 | + (−0.951 − 0.309i)9-s + (−1.76 + 0.896i)13-s + (0.896 − 0.142i)17-s + (1.11 + 1.53i)29-s + (0.809 + 1.58i)37-s + (−0.363 + 1.11i)41-s + i·49-s + (−0.309 − 0.0489i)53-s + (0.363 + 1.11i)61-s + (−0.142 + 0.278i)73-s + (0.809 + 0.587i)81-s + (−1.80 + 0.587i)89-s + (0.278 − 1.76i)97-s − 0.618·101-s + (−1.53 − 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0941 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0941 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8524204791\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8524204791\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (0.951 + 0.309i)T^{2} \) |
| 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 13 | \( 1 + (1.76 - 0.896i)T + (0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.896 + 0.142i)T + (0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 - 0.809i)T^{2} \) |
| 29 | \( 1 + (-1.11 - 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 - 1.58i)T + (-0.587 + 0.809i)T^{2} \) |
| 41 | \( 1 + (0.363 - 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.0489i)T + (0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 + 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.363 - 1.11i)T + (-0.809 + 0.587i)T^{2} \) |
| 67 | \( 1 + (0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 73 | \( 1 + (0.142 - 0.278i)T + (-0.587 - 0.809i)T^{2} \) |
| 79 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 89 | \( 1 + (1.80 - 0.587i)T + (0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (-0.278 + 1.76i)T + (-0.951 - 0.309i)T^{2} \) |
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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
−8.727632290513133356239263873421, −8.146103572199910578698194573441, −7.26112129689592523098074476606, −6.68804698416045192976207335732, −5.82775090448766577181918602843, −4.99205768008191945147831302792, −4.43311906660818571780168113229, −3.12291237010837719250098840176, −2.67601069272635916747390973873, −1.33089168249077969050786231808,
0.48071580671073338346511777072, 2.23057068338761059646618022487, 2.79372034512748355614855077828, 3.79701417062546298362384664642, 4.87006859435132667612283396699, 5.44108096429349854330290763003, 6.10486807889811791485659495572, 7.14191486344257992736525376526, 7.82624745702026319644781447997, 8.255250703109015014772193500180