L(s) = 1 | + 1.41·3-s − 4-s + i·5-s + 1.00·9-s − 1.41·12-s + (0.707 − 0.707i)13-s + 1.41i·15-s + 16-s + 1.41·17-s + 2i·19-s − i·20-s − 1.41·23-s − 25-s − i·31-s − 1.00·36-s + 1.41i·37-s + ⋯ |
L(s) = 1 | + 1.41·3-s − 4-s + i·5-s + 1.00·9-s − 1.41·12-s + (0.707 − 0.707i)13-s + 1.41i·15-s + 16-s + 1.41·17-s + 2i·19-s − i·20-s − 1.41·23-s − 25-s − i·31-s − 1.00·36-s + 1.41i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.529637734\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.529637734\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - iT \) |
| 13 | \( 1 + (-0.707 + 0.707i)T \) |
| 31 | \( 1 + iT \) |
good | 2 | \( 1 + T^{2} \) |
| 3 | \( 1 - 1.41T + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 - 1.41T + T^{2} \) |
| 19 | \( 1 - 2iT - T^{2} \) |
| 23 | \( 1 + 1.41T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 37 | \( 1 - 1.41iT - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - 1.41T + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.41T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.41iT - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−9.526048227715514060879179391755, −8.454569638432037948416064631025, −7.905929419800200784613826461383, −7.67248619175886183864425318000, −6.10943406679652768348230402057, −5.62668873291195176499693562043, −4.04180061169350097558750431173, −3.62453946667223181540704568407, −2.89908403831764470521292838867, −1.61494921427731080941020762608,
1.11190893893154543913427395402, 2.38982735964998717767716201328, 3.61139618957688211556248202949, 4.12786225794346189229918619943, 5.04812650819212702495375033823, 5.88231500710522946659600845852, 7.25004716080695604846422553132, 8.021348505173509776035545632566, 8.587365153684021298219123018196, 9.175789832161759726553532010418