L(s) = 1 | + (−0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s − 11-s + (−0.707 − 0.707i)13-s − 1.00i·15-s + (−0.707 + 0.707i)17-s − 1.41i·19-s + 1.00i·25-s + (−0.707 + 0.707i)27-s − i·29-s − 1.41·31-s + (0.707 + 0.707i)33-s + 1.00i·39-s − 1.41·41-s + (1 + i)43-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)3-s + (0.707 + 0.707i)5-s − 11-s + (−0.707 − 0.707i)13-s − 1.00i·15-s + (−0.707 + 0.707i)17-s − 1.41i·19-s + 1.00i·25-s + (−0.707 + 0.707i)27-s − i·29-s − 1.41·31-s + (0.707 + 0.707i)33-s + 1.00i·39-s − 1.41·41-s + (1 + i)43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3512931725\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3512931725\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 11 | \( 1 + T + T^{2} \) |
| 13 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 17 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + iT - T^{2} \) |
| 31 | \( 1 + 1.41T + T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + 1.41T + T^{2} \) |
| 43 | \( 1 + (-1 - i)T + iT^{2} \) |
| 47 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 53 | \( 1 + (1 + i)T + iT^{2} \) |
| 59 | \( 1 + 1.41iT - T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-1 + i)T - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + iT - T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.707 + 0.707i)T - iT^{2} \) |
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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
−8.054269713171839793250046372209, −7.46839547086896894844041419544, −6.67845427066865322106308724682, −6.26208162057582071890118250352, −5.40074117355870009195026048755, −4.85005289158952231455538026997, −3.46277645114170231807544365058, −2.58425928687734926923274575665, −1.79376110546668870700339771282, −0.19612993140155818150179478129,
1.68873725722883116319785301532, 2.52095276270662202697734115119, 3.86460158089813723806330519505, 4.67375108574372387670404276091, 5.32685672007069472138041500314, 5.62980644947026202755726304271, 6.71396503819196817114274869141, 7.50086704725895216865840540889, 8.368236050098798223527746525436, 9.127535509279268549226747990089