L(s) = 1 | + 3-s − 5-s − 4.91·7-s + 9-s − 4.91·11-s + 13-s − 15-s − 4.33·17-s − 2.57·19-s − 4.91·21-s + 6.33·23-s + 25-s + 27-s + 6·29-s − 1.42·31-s − 4.91·33-s + 4.91·35-s + 9.49·37-s + 39-s + 4.33·41-s + 1.15·43-s − 45-s + 5.42·47-s + 17.1·49-s − 4.33·51-s − 0.338·53-s + 4.91·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.85·7-s + 0.333·9-s − 1.48·11-s + 0.277·13-s − 0.258·15-s − 1.05·17-s − 0.591·19-s − 1.07·21-s + 1.32·23-s + 0.200·25-s + 0.192·27-s + 1.11·29-s − 0.255·31-s − 0.855·33-s + 0.831·35-s + 1.56·37-s + 0.160·39-s + 0.677·41-s + 0.176·43-s − 0.149·45-s + 0.790·47-s + 2.45·49-s − 0.607·51-s − 0.0464·53-s + 0.662·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.152536016\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.152536016\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 + 4.91T + 7T^{2} \) |
| 11 | \( 1 + 4.91T + 11T^{2} \) |
| 17 | \( 1 + 4.33T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 - 6.33T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 1.42T + 31T^{2} \) |
| 37 | \( 1 - 9.49T + 37T^{2} \) |
| 41 | \( 1 - 4.33T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 + 0.338T + 53T^{2} \) |
| 59 | \( 1 - 11.2T + 59T^{2} \) |
| 61 | \( 1 + 10.1T + 61T^{2} \) |
| 67 | \( 1 - 7.25T + 67T^{2} \) |
| 71 | \( 1 + 0.916T + 71T^{2} \) |
| 73 | \( 1 + 3.15T + 73T^{2} \) |
| 79 | \( 1 - 3.49T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 0.338T + 89T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−8.786035946020042312722546565591, −7.971830511855868230321472726576, −7.16095273234630354814137336318, −6.57300077115648274002320638326, −5.77360541882474378872534421570, −4.66824120053552995048248855056, −3.86922971270337248606651619421, −2.86065649458417664913200029732, −2.55918212345189131286272712587, −0.59468445627953642683902386555,
0.59468445627953642683902386555, 2.55918212345189131286272712587, 2.86065649458417664913200029732, 3.86922971270337248606651619421, 4.66824120053552995048248855056, 5.77360541882474378872534421570, 6.57300077115648274002320638326, 7.16095273234630354814137336318, 7.971830511855868230321472726576, 8.786035946020042312722546565591