L(s) = 1 | + 0.449·5-s + 2.04·7-s − 3.46·11-s − 4.79·25-s − 9.34·29-s + 3.60·31-s + 0.921·35-s − 2.79·49-s − 7.55·53-s − 1.55·55-s − 11.3·59-s + 9.79·73-s − 7.10·77-s − 17.4·79-s + 17.3·83-s − 2·97-s − 16.4·101-s − 0.492·103-s − 11.3·107-s + ⋯ |
L(s) = 1 | + 0.201·5-s + 0.774·7-s − 1.04·11-s − 0.959·25-s − 1.73·29-s + 0.647·31-s + 0.155·35-s − 0.399·49-s − 1.03·53-s − 0.209·55-s − 1.47·59-s + 1.14·73-s − 0.809·77-s − 1.96·79-s + 1.90·83-s − 0.203·97-s − 1.63·101-s − 0.0485·103-s − 1.09·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 - 0.449T + 5T^{2} \) |
| 7 | \( 1 - 2.04T + 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 9.34T + 29T^{2} \) |
| 31 | \( 1 - 3.60T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 7.55T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.79T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 - 17.3T + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−7.85152472177745815349508385491, −7.49839393824778111293602714672, −6.43592784499019734119024227716, −5.66102023336549444773855201943, −5.06093811037283114694691485451, −4.28365057850080691351707872323, −3.30984004833142965690932655982, −2.32338197298546205039994443979, −1.52643768638780490220007532166, 0,
1.52643768638780490220007532166, 2.32338197298546205039994443979, 3.30984004833142965690932655982, 4.28365057850080691351707872323, 5.06093811037283114694691485451, 5.66102023336549444773855201943, 6.43592784499019734119024227716, 7.49839393824778111293602714672, 7.85152472177745815349508385491