L(s) = 1 | + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (0.707 + 0.292i)11-s + (−1 + i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.292i)29-s + (0.707 − 1.70i)37-s + (1.70 + 0.707i)43-s + 1.00i·49-s + (1.70 + 0.707i)53-s + 1.00·63-s + (−1.70 + 0.707i)67-s + (0.292 + 0.707i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)7-s + (0.707 − 0.707i)9-s + (0.707 + 0.292i)11-s + (−1 + i)23-s + (−0.707 − 0.707i)25-s + (−0.707 + 0.292i)29-s + (0.707 − 1.70i)37-s + (1.70 + 0.707i)43-s + 1.00i·49-s + (1.70 + 0.707i)53-s + 1.00·63-s + (−1.70 + 0.707i)67-s + (0.292 + 0.707i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.322060299\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.322060299\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 5 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 13 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (1 - i)T - iT^{2} \) |
| 29 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 67 | \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 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
−9.321567562490702486880676241986, −8.946176278037502947939980122962, −7.77688573400485519565483686761, −7.30184504677746441616279397118, −6.12566235568468787622070592799, −5.64521292466912601895852583519, −4.35406246107934631704726748596, −3.86133822578618242995045437430, −2.40960796835518361618559302744, −1.41513987497040380049030504688,
1.28299193648057815971937940009, 2.32250155642156792449899178178, 3.85460609620413376151389506013, 4.34433829279644104583674958906, 5.33239588157186812095336307573, 6.30294304954170728983797977423, 7.21169908918584646357283310710, 7.83001927513730427961286916804, 8.535406472337785840626655447806, 9.536714178140544423416125788511