L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s − i·9-s − 2·11-s − 1.00i·14-s − 1.00·16-s + (−0.707 − 0.707i)18-s + (−1.41 + 1.41i)22-s + (1.41 + 1.41i)23-s + (−0.707 − 0.707i)28-s + (−0.707 + 0.707i)32-s − 1.00·36-s + (1.41 − 1.41i)37-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s − i·9-s − 2·11-s − 1.00i·14-s − 1.00·16-s + (−0.707 − 0.707i)18-s + (−1.41 + 1.41i)22-s + (1.41 + 1.41i)23-s + (−0.707 − 0.707i)28-s + (−0.707 + 0.707i)32-s − 1.00·36-s + (1.41 − 1.41i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.454543453\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.454543453\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + iT^{2} \) |
| 11 | \( 1 + 2T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (-1.41 + 1.41i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.691748151295090939711516287963, −8.888717788292081697757836396209, −7.70120317507425053881318247578, −7.10921833043693498197934647900, −5.83622774236759075878055463790, −5.22282530444739120148876087949, −4.32983848716210785356525031360, −3.36824504198896216897302967310, −2.43216942156901137481886955249, −0.990213845625907065929441093178,
2.36079290218657648514300410497, 2.86991368829625827495916447409, 4.55386295984599860582520535068, 5.05297713927023676042057602166, 5.63856108595301854328853032195, 6.76176677786536511397659401339, 7.73798201724309262489502264937, 8.171039605770033485512851782016, 8.812836785592723978767507124967, 10.15086681854757701546933941973