L(s) = 1 | − 2·5-s + 2·11-s − 4·13-s + 19-s + 8·23-s − 25-s + 2·29-s + 2·31-s − 8·37-s + 2·41-s − 4·43-s − 4·47-s − 7·49-s − 2·53-s − 4·55-s − 10·61-s + 8·65-s − 16·71-s + 6·73-s − 14·79-s − 6·83-s + 18·89-s − 2·95-s + 10·97-s − 14·101-s − 6·103-s + 8·107-s + ⋯ |
L(s) = 1 | − 0.894·5-s + 0.603·11-s − 1.10·13-s + 0.229·19-s + 1.66·23-s − 1/5·25-s + 0.371·29-s + 0.359·31-s − 1.31·37-s + 0.312·41-s − 0.609·43-s − 0.583·47-s − 49-s − 0.274·53-s − 0.539·55-s − 1.28·61-s + 0.992·65-s − 1.89·71-s + 0.702·73-s − 1.57·79-s − 0.658·83-s + 1.90·89-s − 0.205·95-s + 1.01·97-s − 1.39·101-s − 0.591·103-s + 0.773·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.452227845122068783884853748207, −7.58013492860400969550788127744, −7.08620400459534305912068044575, −6.28130026977410340618982546917, −5.10534376057806946560314487461, −4.57679173230598028958874190152, −3.55941614958292228965944271223, −2.80711203445431957541164382037, −1.43340605848042057325894030833, 0,
1.43340605848042057325894030833, 2.80711203445431957541164382037, 3.55941614958292228965944271223, 4.57679173230598028958874190152, 5.10534376057806946560314487461, 6.28130026977410340618982546917, 7.08620400459534305912068044575, 7.58013492860400969550788127744, 8.452227845122068783884853748207