L(s) = 1 | + (−0.987 − 0.156i)5-s + (−0.278 − 0.142i)13-s + (0.0966 − 0.610i)17-s + (0.951 + 0.309i)25-s + (−1.44 − 1.04i)29-s + (0.809 − 1.58i)37-s + (−0.297 + 0.0966i)41-s + i·49-s + (−0.253 − 1.59i)53-s + (0.363 − 1.11i)61-s + (0.253 + 0.183i)65-s + (−0.896 − 1.76i)73-s + (−0.190 + 0.587i)85-s + (0.280 − 0.863i)89-s + (−0.142 − 0.896i)97-s + ⋯ |
L(s) = 1 | + (−0.987 − 0.156i)5-s + (−0.278 − 0.142i)13-s + (0.0966 − 0.610i)17-s + (0.951 + 0.309i)25-s + (−1.44 − 1.04i)29-s + (0.809 − 1.58i)37-s + (−0.297 + 0.0966i)41-s + i·49-s + (−0.253 − 1.59i)53-s + (0.363 − 1.11i)61-s + (0.253 + 0.183i)65-s + (−0.896 − 1.76i)73-s + (−0.190 + 0.587i)85-s + (0.280 − 0.863i)89-s + (−0.142 − 0.896i)97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0755 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0755 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7430454531\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7430454531\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (0.987 + 0.156i)T \) |
good | 7 | \( 1 - iT^{2} \) |
| 11 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 13 | \( 1 + (0.278 + 0.142i)T + (0.587 + 0.809i)T^{2} \) |
| 17 | \( 1 + (-0.0966 + 0.610i)T + (-0.951 - 0.309i)T^{2} \) |
| 19 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + (-0.587 + 0.809i)T^{2} \) |
| 29 | \( 1 + (1.44 + 1.04i)T + (0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 37 | \( 1 + (-0.809 + 1.58i)T + (-0.587 - 0.809i)T^{2} \) |
| 41 | \( 1 + (0.297 - 0.0966i)T + (0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + (-0.951 + 0.309i)T^{2} \) |
| 53 | \( 1 + (0.253 + 1.59i)T + (-0.951 + 0.309i)T^{2} \) |
| 59 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 71 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (0.896 + 1.76i)T + (-0.587 + 0.809i)T^{2} \) |
| 79 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 83 | \( 1 + (-0.951 - 0.309i)T^{2} \) |
| 89 | \( 1 + (-0.280 + 0.863i)T + (-0.809 - 0.587i)T^{2} \) |
| 97 | \( 1 + (0.142 + 0.896i)T + (-0.951 + 0.309i)T^{2} \) |
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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
−8.460218039300146511825745884632, −7.64941602002354110735172196734, −7.35525352678039101539338698089, −6.32841374638208204992273631555, −5.46478795303600990920865778168, −4.64999906319853145585410837951, −3.90296310367406038392761341429, −3.10068739523777606236889968626, −2.00055932447720786118785168726, −0.45349841264741748273803302251,
1.33933212865558841797143027348, 2.64717413404121684464958048383, 3.54088921779989314674209229632, 4.23814369628536458451558193175, 5.07480750024951628182017614249, 5.96218533745321343031563332292, 6.88820304098573321596588758896, 7.41557188784808397408226602159, 8.198588336341042956097854367279, 8.761224958195181493898539922285