L(s) = 1 | + (0.951 − 0.309i)9-s + (−0.896 + 1.76i)13-s + (−0.142 + 0.896i)17-s + (−1.11 + 1.53i)29-s + (1.58 + 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (0.0489 + 0.309i)53-s + (0.363 − 1.11i)61-s + (−0.278 + 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (−1.76 + 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)9-s + (−0.896 + 1.76i)13-s + (−0.142 + 0.896i)17-s + (−1.11 + 1.53i)29-s + (1.58 + 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (0.0489 + 0.309i)53-s + (0.363 − 1.11i)61-s + (−0.278 + 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (−1.76 + 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.215523878\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.215523878\) |
\(L(1)\) |
|
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 | \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \) |
| 17 | \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \) |
| 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 | \( 1 + (-1.58 - 0.809i)T + (0.587 + 0.809i)T^{2} \) |
| 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 | \( 1 + (-0.0489 - 0.309i)T + (-0.951 + 0.309i)T^{2} \) |
| 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 | \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \) |
| 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 | \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \) |
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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
−8.936109210704332995079557041838, −7.946992168947349799146559467082, −7.13967258695315259701561422407, −6.74219191672740514947373011864, −5.88875441566962487467539298371, −4.81898593312863211863415280434, −4.26487735037245109466832037990, −3.48444567854195544248087335568, −2.19462611321061363461237086159, −1.45272634367953630068183288294,
0.69952285132200704004475906738, 2.14485206147288123826099364059, 2.90822105014798316506708198786, 3.96557487189154611714231957546, 4.78257352487981085102458327090, 5.45352889028112300227393529523, 6.24415856981410485079688219314, 7.34582093932608177743243266760, 7.59578698862418475996734537593, 8.325951245511317579203344336193