L(s) = 1 | + (0.951 + 0.309i)9-s + (0.142 + 0.278i)13-s + (−0.278 − 1.76i)17-s + (1.11 + 1.53i)29-s + (0.809 − 0.412i)37-s + (0.363 − 1.11i)41-s − i·49-s + (−0.309 + 1.95i)53-s + (−0.363 − 1.11i)61-s + (1.76 + 0.896i)73-s + (0.809 + 0.587i)81-s + (−1.80 + 0.587i)89-s + (−0.896 − 0.142i)97-s − 0.618·101-s + (1.53 + 0.5i)109-s + ⋯ |
L(s) = 1 | + (0.951 + 0.309i)9-s + (0.142 + 0.278i)13-s + (−0.278 − 1.76i)17-s + (1.11 + 1.53i)29-s + (0.809 − 0.412i)37-s + (0.363 − 1.11i)41-s − i·49-s + (−0.309 + 1.95i)53-s + (−0.363 − 1.11i)61-s + (1.76 + 0.896i)73-s + (0.809 + 0.587i)81-s + (−1.80 + 0.587i)89-s + (−0.896 − 0.142i)97-s − 0.618·101-s + (1.53 + 0.5i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.433158447\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.433158447\) |
\(L(1)\) |
|
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 | \( 1 + (-0.142 - 0.278i)T + (-0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (0.278 + 1.76i)T + (-0.951 + 0.309i)T^{2} \) |
| 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 | \( 1 + (-0.809 + 0.412i)T + (0.587 - 0.809i)T^{2} \) |
| 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 | \( 1 + (0.309 - 1.95i)T + (-0.951 - 0.309i)T^{2} \) |
| 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 | \( 1 + (-1.76 - 0.896i)T + (0.587 + 0.809i)T^{2} \) |
| 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 | \( 1 + (0.896 + 0.142i)T + (0.951 + 0.309i)T^{2} \) |
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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
−8.684177896088340806778709659554, −7.77813304337896889801104315465, −7.07696883134046784489884065113, −6.67102674378851092570350481336, −5.51865993930006968129874883538, −4.82555032191428204069120998304, −4.16855084994953240017632508608, −3.10644174595191372097002541949, −2.21338227684567102175597901696, −1.03929345663696397723061347531,
1.12404826424890290748201183915, 2.14314490662034437229156457093, 3.26671263641733394939534914498, 4.17739233530411504358814569038, 4.65057092061303244437634668960, 5.92860275214370784695192142568, 6.32329636484908448427440554272, 7.14471488424670489859552369971, 8.124848606683721861021854230033, 8.356189308896358762033099074657