| L(s) = 1 | − i·4-s + (0.707 + 0.707i)5-s + (−1 − i)7-s − 1.41i·11-s − 16-s − i·19-s + (0.707 − 0.707i)20-s + (1.41 + 1.41i)23-s + 1.00i·25-s + (−1 + i)28-s − 1.41i·35-s + (1 − i)43-s − 1.41·44-s + (−1.41 + 1.41i)47-s + i·49-s + ⋯ |
| L(s) = 1 | − i·4-s + (0.707 + 0.707i)5-s + (−1 − i)7-s − 1.41i·11-s − 16-s − i·19-s + (0.707 − 0.707i)20-s + (1.41 + 1.41i)23-s + 1.00i·25-s + (−1 + i)28-s − 1.41i·35-s + (1 − i)43-s − 1.41·44-s + (−1.41 + 1.41i)47-s + i·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9755341045\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9755341045\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.707 - 0.707i)T \) |
| 19 | \( 1 + iT \) |
| good | 2 | \( 1 + iT^{2} \) |
| 7 | \( 1 + (1 + i)T + iT^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 23 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + (-1 + i)T - iT^{2} \) |
| 47 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 2T + T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1 - i)T - iT^{2} \) |
| 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
−10.19574699429452506990329368258, −9.547396638742537849976681641996, −8.873644172257533771713249443360, −7.31780188478307991858998662991, −6.69683490653387019619337918651, −5.95762773566764123329529615527, −5.14362758230769134078721484760, −3.64900819978712373072822762962, −2.75184923333917728736360534650, −1.04879198636909619473240441296,
2.07441109748410964014599822809, 2.98062933772815863863763820858, 4.29713547017525582512552863543, 5.19409813399229753798455324267, 6.31225992798739807931633303070, 7.02905413148285075207399633218, 8.199622589531863281727206494030, 8.908905916114799719742790665966, 9.563061340608739385618638598254, 10.27070526185304782205218431523
