| L(s) = 1 | − i·3-s − 3.82i·7-s − 9-s − 5.82i·11-s − 4.82·13-s + (−3 + 2.82i)17-s + 3·19-s − 3.82·21-s − 8.48i·23-s + i·27-s − 4.17i·29-s + 6.82i·31-s − 5.82·33-s − 9.48i·37-s + 4.82i·39-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.44i·7-s − 0.333·9-s − 1.75i·11-s − 1.33·13-s + (−0.727 + 0.685i)17-s + 0.688·19-s − 0.835·21-s − 1.76i·23-s + 0.192i·27-s − 0.774i·29-s + 1.22i·31-s − 1.01·33-s − 1.55i·37-s + 0.773i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8433980795\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8433980795\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 + (3 - 2.82i)T \) |
| good | 7 | \( 1 + 3.82iT - 7T^{2} \) |
| 11 | \( 1 + 5.82iT - 11T^{2} \) |
| 13 | \( 1 + 4.82T + 13T^{2} \) |
| 19 | \( 1 - 3T + 19T^{2} \) |
| 23 | \( 1 + 8.48iT - 23T^{2} \) |
| 29 | \( 1 + 4.17iT - 29T^{2} \) |
| 31 | \( 1 - 6.82iT - 31T^{2} \) |
| 37 | \( 1 + 9.48iT - 37T^{2} \) |
| 41 | \( 1 - 5.82iT - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 8.65T + 47T^{2} \) |
| 53 | \( 1 + 8.31T + 53T^{2} \) |
| 59 | \( 1 - 5.17T + 59T^{2} \) |
| 61 | \( 1 + 8.48iT - 61T^{2} \) |
| 67 | \( 1 - 2.48T + 67T^{2} \) |
| 71 | \( 1 - 9.65iT - 71T^{2} \) |
| 73 | \( 1 - 15.8iT - 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 + 14iT - 97T^{2} \) |
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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
−7.80158835853701633578330296200, −6.96366531313914657836075417730, −6.55776238432552593114735720002, −5.68861419248333265588312129303, −4.78773684820633391484660174386, −3.99933371845775111579920435011, −3.16408091320606874755222308049, −2.27834903813108712766594745841, −0.982879772962910504069128045494, −0.24844650271751809390083440407,
1.83248774923960152362474812736, 2.45851777243875250056232079648, 3.30664110855255162509761052091, 4.48447169963053718285370602680, 5.06049925478264888747912194188, 5.47901560000537287901638044339, 6.55909877820798087556049710670, 7.34133568833652955174180797408, 7.83735000198691900018708947660, 8.974395016894915966271302631378
